Soliton Solutions of Noncommutative Integrable Systems

Transcription

1 Soliton Solutions of Noncommutative Integrable Systems by Craig M Sooman A thesis submitted to the Faculty of Information and Mathematical Sciences at the University of Glasgow for the degree of Doctor of Philosophy September 009 c C M Sooman 009

2 1 This thesis is concerned with solutions of noncommutative integrable systems where the noncommutativity arises through the dependent variables in either the hierarchy or Lax pair generating the equation Both Chapters 1 and are entirely made up of background material and contain no new material Furthermore, these chapters are concerned with commutative equations Chapter 1 outlines some of the basic concepts of integrable systems including historical attempts at finding solutions of the KdV equation, the Lax method and Hirota s direct method for finding multi-soliton solutions of an integrable system Chapter extends the ideas in Chapter 1 from equations of one spatial dimension to equations of two spatial dimensions, namely the KP and mkp equations Chapter also covers the concepts of hierarchies and Darboux transformations The Darboux transformations are iterated to give multi-soliton solutions of the KP and mkp equations Furthermore, this chapter shows that multi-soliton solutions can be expressed as two types of determinant: the Wronskian and the Grammian These determinantal solutions are then verified directly In Chapter 3, the ideas detailed in the preceding chapters are extended to the noncommutative setting We begin by outlining some known material on quasideterminants, a noncommutative KP hierarchy containing a noncommutative KP equation, and also two families of solutions The two families of solutions are obtained from Darboux transformations and can be expressed as quasideterminants One family of solutions is termed quasiwronskian and the other quasigrammian as both reduce to Wronskian and Grammian determinants when their entries commute Both families of solutions are then verified directly The remainder of Chapter 3 is original material, based on joint work with Claire Gilson and Jon Nimmo Building on some known results, the solutions obtained from the Darboux transformations are specified as matrices These solutions have interesting interaction properties not found in the commutative setting We therefore show various plots of the solutions illustrating these properties In Chapter 4, we repeat all of the work of Chapter 3 for a noncommutative mkp equation The material in this chapter is again based on joint work with Claire Gilson and Jon Nimmo and is mainly original The original material in Chapters 3 and 4 appears in [0] and in [1] Chapter 5 builds on the work of Chapters 3 and 4 and is concerned with exponentially localised structures called dromions, which are obtained by taking the determinant of the matrix solutions of the noncommutative KP and mkp equations For both equations, we

3 look at a three-dromion structure from which we then perform a detailed asymptotic analysis This aymptotic forms show interesting interaction properties which are demonstrated by various plots This chapter is entirely the author s own work Chapter 6 presents a summary and conclusions of the thesis

4 3 Firstly, I would like to thank my supervisors, Dr Claire Gilson and Dr Jon Nimmo, for their support and encouragement shown to me throughout the development of this thesis Thanks are also due to the Engineering and Physical Sciences Research Council for the financial support which enabled me to undertake the research required to compile this thesis There are several people in the department of mathematics at the University of Glasgow who made the department a pleasant and enjoyable place in which to learn The list of such people is exhaustive, but I would particularly like to express my gratefulness to the janitors, fellow postgraduates, academic staff, secretarial staff and several undergraduates whom I enjoyed tutoring I would also like to thank both my mother and grandmother for their constant support and assistance during my time as an undergraduate and postgraduate at the University of Glasgow Finally, I must thank Monica, who for the last four and a half years, has been a continuous source of encouragement, support and friendship Craig M Sooman Edinburgh to York 10th January 010

5 Contents 1 Introduction 7 11 Sir John Scott Russell s observation of a solitary wave 7 1 The Korteweg-de Vries equation 8 13 The Lax method 9 14 The modified KdV equation Hirota s direct method Summary 14 The KP and mkp equations 16 1 The Kadomtsev-Petviashvili equation The KP hierarchy 17 1 Wronskian solutions obtained from Hirota s method Wronskian solutions obtained from Darboux transformations 0 14 Grammian solutions obtained from Hirota s method 4 15 Grammian solutions obtained from binary Darboux transformations 5 The modified Kadomstsev-Petviashvili equation 9 1 The mkp hierarchy 9 Wronskian solutions obtained from Hirota s method 30 3 Wronskian solutions obtained from Darboux transformations 31 4 Grammian solutions obtained from Hirota s method 33 5 Grammian solutions obtained from Darboux transformations 34 3 Direct verification of the solutions Derivatives of Wronskian determinants 36 3 Laplace expansion of determinants Plücker relations 39 4

6 CONTENTS 5 34 Derivatives of Grammian determinants Jacobi identity for determinants 41 4 The Miura transformation 4 3 A noncommutative KP equation Quasideterminants The matrix inverse Noncommutative Jacobi identity Elementary row and column operations Comparison with commutative determinants Quasi-Plücker coordinates 49 3 A noncommutative KP hierarchy Quasiwronskian solutions obtained from Darboux transformations Quasigrammian solutions obtained from binary Darboux transformations 5 35 Reduction to commutative Wronskian and Grammian solutions Direct verification of the solutions Derivatives of quasideterminants Derivatives of quasiwronskians Derivatives of quasigrammians Matrix solutions Plots of the matrix solutions Reduction to matrix KdV 73 4 A noncommutative mkp equation A noncommutative mkp hierarchy 75 4 Quasiwronskian solutions obtained from Darboux transformations Quasigrammian solutions obtained from binary Darboux transformations Reduction to commutative Wronskian and Grammian solutions Direct verification of the solutions Matrix solutions 9 47 Plots of the matrix solutions The noncommutative Miura transformation 103

7 CONTENTS 6 5 Dromions of the matrix equations Matrix KP single dromion A three-dromion example Summary of the three-dromion structure Plots of dromions 1 5 Matrix mkp single dromion A three-dromion example 19 5 Summary of the three-dromion structure Plots of dromions Summary and conclusions 140 References 14

8 Chapter 1 Introduction 11 Sir John Scott Russell s observation of a solitary wave The solitary wave can be traced back to 1834 when Sir John Scott Russell observed what he called the great wave of translation on the Union Canal in Scotland Reporting his observation in [47], he described the physical characteristics of a mass of water put in motion by the sudden stoppage of a horse-drawn boat According to Russell, the mass of water rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed Russell then attempted to recreate this phenomenon in laboratory experiments where he created solitary waves by dropping weights at one end of a water channel He then deduced that the speed of a solitary wave, c, is given by c = gh + a, 11 where a denotes amplitude, h is the undisturbed depth of the water and g is the acceleration of gravity From equation 11 it is clear that taller waves travel faster Continuing on a mathematical theme, Boussinesq 1871 and Lord Rayleigh 1876, used 11 in order to find an expression for the wave profile They showed that the wave profile z = ζx, t is given by ζx, t = a sech βx ct, 1 7

9 CHAPTER 1 INTRODUCTION 8 where for a > 0 and a h a, wavelength 1 β 3a β = 4h h + a << 1 Equation 1 represents a right-travelling wave with amplitude and speed c 1 The Korteweg-de Vries equation Despite being able to derive 1, neither Boussinesq or Lord Rayleigh managed to find an equation governing this wave profile However, in 1895, Korteweg and de Vries were successful in finding a mathematical schematization to describe Russell s observation They developed a nonlinear partial differential equation which governs one-dimensional waves The celebrated Korteweg-de Vries [6] normally abbreviated to KdV equation is u t + αβ γ u xxx + β γ 3 uu x = 0, 13 where the subscripts denote partial differentiation and α, β and γ are constants Equation 13 is a general form of the KdV equation In this thesis, we will use the following version of the KdV equation: u t + u xxx + 6uu x = 0 14 To find a solitary-wave solution of equation 14, we let u = lx ct = lξ for some constant c > 0 so that our solution is a right-travelling wave Therefore l must satisfy where d dξ Integrating equation 15 once gives cl + 6ll + l = 0, 15 cl + 3l + l = C 1, 16 where C 1 is an arbitrary constant Multiplying both sides of equation 17 by l and integrating again yields c l + l l = C 1 l + C, 17 where C is an arbitrary constant By setting C 1 = C = 0 so that we have a wave which has l 0, l 0 and l 0 as ξ ±, we have the first-order ordinary differential equation l = ±l c l 18

10 CHAPTER 1 INTRODUCTION 9 By making the substitution l = 1 sech θ, equation 18 leads to the solution ux, t = c c sech x ct x 0, 19 where the choice ± has been eliminated since the solution is an even function, and x 0 is an arbitrary constant In fact, x 0 plays an important role in the behaviour of the solution: it is the phase constant the position of the peak of the wave at t = 0 Some time passed before the KdV equation was shown to possess multi-soliton solutions In 1965, Zabusky and Kruskal [33] considered the initial-value problem for a version of the KdV equation u t + uu x + δ u xxx = They solved this equation with ux, 0 = cosπx, 0 x and u, u x, u xxx periodic on [0, ] Their results showed that the initial profile seperated into eight sech -like functions propagating around the system with different speeds The sech -like functions collided but emerged from interaction with all of their characteristics preserved This is as a result of the balancing of the nonlinear and dispersive terms in the equation Owing to these particle-like properties, Zabusky and Kruskal termed the solitary-wave solution a soliton, where the suffix -on indicates a particle Whilst no precise mathematical definition of the soliton exists, Drazin and Johnson define solitons in [9] as any solution of a nonlinear equation or system which: 1 represents a wave of permanent form; is localised, so that it decays or approaches a constant at infinity; 3 can interact strongly with other solitons and maintain its identity 13 The Lax method In 1968, Lax presented a method [35] which represents nonlinear evolution equations with differential operators that are linear in x The work of Lax requires two operators, L and M, which operate on elements of L R, the space of integrable functions on the real line, endowed with an inner product φ, ψ = + φ ψ dx 111

11 CHAPTER 1 INTRODUCTION 10 Both L and M are self-adjoint so that L[φ], ψ = φ, L[ψ] and M[φ], ψ = φ, M[ψ] φ, ψ L R In the spirit of finding exact solutions via inverse scattering, one has the spectral problem L[ψ] = λψ, 11 so that ψ is an eigenfunction for L with eigenvalue λ In addition, the eigenfunction ψ evolves in time according to ψ t = M[ψ] 113 Lax showed that if equations 11 and 113 both hold, then the operators L and M satisfy the relation L t + [L, M] = 0, 114 where [L, M] = LM ML denotes the commutator of L and M To see this, we differentiate both sides of 11 with respect to t and then substitute equation 113 into the resulting equation Doing so gives λ t ψ = L t [ψ] + L[ψ t ] λψ t = L t [ψ] + LM[ψ] M[λψ] = L t [ψ] + LM[ψ] ML[ψ] = L t + [L, M][ψ] 115 Solving equation 115 for nontrivial eigenfunction ψ and choosing λ t = 0 gives equation 114 Since λ t vanishes, every eigenvalue of L is a constant Throughout this thesis, we use a more convenient but equivalent form of 114, by incorporating t into the operator M The KdV equation 13 provides us with a prototypical example of the Lax representation If we choose the operators L KdV = x + u, M KdV = 4 x 3 + 6u x + 3u x + t, 116 then [L KdV, M KdV ][ψ] = u t + u xxx + 6uu x ψ 117 Therefore, we must have that [L KdV, M KdV ] = 0 if and only if u is a solution of the KdV equation When a nonlinear evolution equation can be represented in this way, it is said to have a Lax representation and the two operators used are referred to as a Lax pair

12 CHAPTER 1 INTRODUCTION The modified KdV equation By making a simple modification to the nonlinear term in 13, we obtain the modified KdV abbreviated to mkdv equation w t + w xxx 6w w x = 0, 118 which will play an important role in what follows In 1968, Miura [39] showed that if w satisfies equation 118, then u, defined by u = w + w x, 119 satisfies the KdV equation 13 Substituting 119 into 13 gives w + x w t + w xxx 6w w x = 0 10 Therefore, every solution of the KdV equation 13 can be obtained from a solution of the mkdv equation 118 However, the converse of this statement is false Equation 118 also has a Lax representation: it can be thought of as the compatibility condition of the operators L mkdv = x + w x, M mkdv = 4 3 x + 1w x + 6w x + w x + t Hirota s direct method Hirota proposed the direct method [8] in 1971 Hirota s method transforms an evolution equation into a type of bilinear differential equation via a transformation of the dependent variable From this platform we can find exact solutions In devising this method, Hirota introduced a new differential operator, the D-operator: DxD l t m a b := x l x t m t ax, tbx, t, 1 x=x, t=t

13 CHAPTER 1 INTRODUCTION 1 for nonnegative integers l and m For example D x D t a b = x x t t ax, tbx, t = x x a t b ab t x=x, t=t x=x, t=t = a xt b a x b t a t b x + ab x=x x t, t=t = a xt b a x b t a t b x + ab xt There are many properties of the D-operator that can be called upon to assist in solving differential equations In what follows, the following three properties will be utilised: D x D t a 1 = a xt = D x D t 1 a, 13 Dx 4 a 1 = a xxxx = Dx 4 1 a, 14 Dx m Dt n exp Λ 1 exp Λ = λ 1 λ m λ 3 λ 3 n 1 exp Λ1 + Λ, 15 where Λ i = λ i x λ i t + λ i 0, i = 1, and λ i0 is the phase-constant Exact solutions of the KdV equation can be found using Hirota s direct method The first step is to make the dependent variable transformation u = log τ xx 16 Substituting directly into the KdV equation 14 gives an equation involving τ: τ xt τ τ x τ t + τ xxxx τ 4τ xxx τ x + 3τxx = 0 17 Using the D-operator, we can express equation 17 as D x D t + Dxτ 4 τ = 0 18 To find the solution τ, we expand it as a power series in ɛ << 1: τ = 1 + ɛτ 1 + ɛ τ + ɛ 3 τ Substituting the above equation into 18 and collecting terms in each order of ɛ gives ɛ : D x D t + Dxτ τ 1 = 0, 130 ɛ : D x D t + Dxτ τ 1 τ τ = 0, 131 ɛ 3 : D x D t + Dxτ τ τ 1 + τ 1 τ + 1 τ 3 = 0, 13

14 CHAPTER 1 INTRODUCTION 13 Properties 13 and 14 of the D-operator imply that the coefficient of ɛ 130 is equivalent to x t + 3 x 3 τ 1 = One solution of the above equation is τ 1 = e Λ Using properties 13 and 14 of the D-operator, 131, the coefficient of ɛ, can be written as x t + 3 x 3 τ = D x D t + Dxτ 3 1 τ Upon substitution of 134 into 135, and using property 15 of the D-operator, we obtain x t + 3 x 3 τ = We may choose the solution of equation 136 to be τ = 0 Similar calculations apply for τ n and we may choose τ n = 0, n =, 3,, for all x,t The expansion of τ can therefore be truncated at τ = 1 + ɛτ 1 Substituting this expression for τ into 16 gives u = log 1 + e Λ 1 xx, = 1 1 λ 1 sech Λ 1, 137 in which ɛ has been absorbed into the phase-constant λ 10 Equation 137 represents a travelling wave solution Hirota s method can also be used to find the two-soliton solution of the KdV equation Consider equation 133 Since this equation is linear in τ 1, we may use the linear superposition principle and choose the solution τ 1 = e Λ 1 + e Λ Upon substitution of this choice of τ into the coefficient of ɛ and using property 15, we obtain x t + 3 x 3 τ = λ 1 λ 4 e Λ 1+Λ, 138

15 CHAPTER 1 INTRODUCTION 14 which has solution τ = τ into the coefficient of ɛ 3 gives λ1 λ λ 1 +λ e Λ 1 +Λ Using 13, 14 and substituting τ 1 and x t + 3 x 3 τ 3 = We choose τ 3 = 0 as a solution of 139 and may then choose τ n = 0, n = 3, 4,, for all x, t The expansion of τ can therefore be truncated at 1 + ɛτ 1 + ɛ τ So we have τ = 1 + ɛ e Λ 1 + e Λ + ɛ λ1 λ e Λ 1+Λ 140 λ 1 + λ By substituting 140 into 16 and absorbing ɛ into the phase-constants, the two-soliton solution can be written as in which Hirota showed that if u = log 1 + e Λ 1 + e Λ + a 1 e Λ 1+Λ, 141 xx λ1 λ a 1 = λ 1 + λ λi λ j a ij =, λ i + λ j then by writing a ij = e A ij, the n-soliton solution can be expressed as τ = n i exp ω i Λ i + A ij ω i ω j, n=1 where is the summation over all possible combinations of ω 1 = 0, 1, ω = 0, 1,, ω n = 0, 1 and n i<j is the summation over all possible pairs i, j where i, j {1,,, n} and i < j i<j 16 Summary In this chapter, an historical account of the soliton and its association with the KdV equation was discussed, as well as some elementary ideas from integrable systems which help lay the foundations for the material in this thesis An outline of the construction of soliton solutions obtained from Hirota s method was given and will be referred to in the next chapter for equations in two spatial dimensions Later in this thesis, we shall see that Hirota s method cannot be used to find soliton solutions of noncommutative integrable systems It

16 CHAPTER 1 INTRODUCTION 15 was shown that the KdV equation has a Lax representation and possesses multi-soliton solutions The idea of a Lax representation is very important for future chapters as the Lax pair used is an essential ingredient needed to generate noncommutative equations Furthermore, Lax pairs are heavily used in Darboux transformations, introduced in Chapter, which we later use as an alternative to Hirota s method for obtaining soliton solutions of noncommutative equations We also introduced the mkdv equation and it was shown that the Miura transformation mapped solutions of the mkdv equation to solutions of the KdV equation Most of the original work in this thesis centres around a noncommutative mkp equation, which can be thought of as a generalisation to two spatial dimensions of a noncommutative mkdv equation In this work, in addition to finding soliton solutions, we also replicate the Miura transformation to map solutions of a noncommutative mkp equation to solutions of a noncommutative KP equation

17 Chapter The KP and mkp equations In this chapter, we are concerned with generalisations to two spatial dimensions of the KdV and mkdv equations, which are the Kadomstev Petviashvili KP and modified KP mkp equations respectively Both of these equations are known to have multi-soliton solutions which can be obtained from Hirota s direct method Alternatively, they may be found from Darboux transformations, which we shall visit later in this chapter The solutions can be expressed compactly as determinants and then verified directly The main purpose of this chapter is to demonstrate these methods for both the KP and mkp equations, as the results that we will obtain can be related to noncommutative results in later chapters Let us begin with the KP equation, which serves as a prototypical example 1 The Kadomtsev-Petviashvili equation The KP equation is u t + u xxx + 6uu x x + 3u yy = 0, 1 which can also be written in potential form v t + v xxx + 3vx x + 3v yy = 0, where u = v x Kadomtsev and Petviashvili [31] derived the equation in 1971 and it was subsequently named after them By neglecting the y-derivative term in 1, we recover the KdV equation 14 The Lax pair for the KP equation is L KP = x + v x y, 3 M KP = 4 x 3 + 6v x x + 3v xx + 3v y + t, 4 16

18 CHAPTER THE KP AND MKP EQUATIONS 17 whose compatibility condition [L KP, M KP ] = 0 gives 11 The KP hierarchy The KP hierarchy is an infinite set of nonlinear evolution equations in infinitely many functions u, u 1, u, of the infinitely many variables x 1, x, x 3, There are various approaches to constructing this hierarchy In this thesis, we use the method of Gelfand and Dickii [15] and Sato [5, 48] To construct the hierarchy, we need the following extended version of the Leibnitz rule: xu i = i j u j x j i j x, 5 j 0 for i Z We define the binomial coefficients in 5 to be i = j 1 j = 0 ii 1 i j+1 jj 1 1 j 0 For example, 1 x u = 1 u x = u 1 x u x x 1 u x x u xx 3 x, x u = u x u x x 3 + 3u xx x 4, x 3 u = u x 3 3u x x 4 + 6u xx x 5 1 u xx x 3 + Construction of the KP hierarchy also requires the use of a pseudodifferential operator L = i Z u i a i x, of order a Associated with this pseudodifferential operator we consider natural projections P k, such that P k L = i k u i i x For the KP hierarchy, we use the pseudodifferential operator L KP = x + 1 u 1 x + u x Let L = L KP Then the KP hierarchy is defined to be + u 3 3 x + 6 L xq = [P 0 L q, L], q = 1,, 3, 7

19 CHAPTER THE KP AND MKP EQUATIONS 18 In general, the operators P 0 L q will be differential operators of order q associated with the fields u, u,, u q 1 The first three natural projections P 0 L q are P 0 L = x, P 0 L = x + u, P 0 L 3 = x u x + 3 u x + 3u Thus, the evolution equation 7 gives u x1 = u x, u x1 = u x, L x1 = [P 0 L, L] u 3x1 = u 3x,, 8 L x = [P 0 L, L] L x3 = [P 0 L 3, L] u y = u xx + 4u x, u y = u xx + u 3x + 1 uu x, u 3y = u 3xx + u 4x 1 uu xx + u u x,, u t = u xxx + 3uu x + 6u xx + 6u 3x, u t = u xxx + 3uu x + 3u 3xx + 3u 4x,, 9 10 where we have set x 1 = x, x = y and x 3 = t These are the first three equations of the KP hierarchy Other equations for L xq = [P 0 L q, L] may also be considered for q = 4, 5, By integrating with respect to x, we can recursively express the fields u, u 3, in terms of u and its x- and y-derivatives The fields u, u 3, can then be eliminated through 9 allowing us to rewrite the first component in 10 in terms of u and its x- and y-derivatives The equation we obtain is the KP equation 1, where the scaling t 4t has been made 1 Wronskian solutions obtained from Hirota s method When working in two spatial dimensions, the D-operator is defined by DxD l y m Dt n a b 11 = x l x y m y t n t ax, y, tbx, y, t, x=x, y=y, t=t

20 CHAPTER THE KP AND MKP EQUATIONS 19 for nonnegative integers l, m and n The dependent variable transformation for the KP equation is u = log τ xx 1 Substituting this into the KP equation 1 gives the bilinear form of the KP equation: ττ xt τ x τ t + ττ xxxx + 3τ xx 4τ x τ xxx + 3ττ yy 3τ y = 0, 13 D 4 x + D x D t + 3D yτ τ = 0 14 Soliton solutions are obtained by using the same perturbation method outlined in Chapter 1 For the one-soliton solution, we truncate the series at τ = 1 + ɛτ 1, so that τ 1 = 1 + e Λ 1, where Λ 1 = η 1 γ 1 + Λ 10, η 1 = p 1 x + p 1 y 4p 1 t + η 1 0, γ 1 = q 1 x + q 1 y 4q 1 t + γ 1 0, p 1, q 1, η 10, γ 10 are constants and Λ 10 is the phase-constant Then we have the one-soliton solution u = log1 + e Λ 1 xx, = 1 p 1 q 1 sech 1 Λ 1 0, where ɛ has been absorbed into the phase-constant Λ 10 For the two-soliton solution, we obtain Let us now define the functions τ = 1 + e Λ 1 + e Λ + p 1 p q 1 q p 1 q q 1 p eλ 1+Λ 15 θ i = e η i + e γ i, in which η i = p i x + p i y 4p i t + η 1 0, γ i = q i x + q i y 4q i t + γ 1 0 and p i, q i, η 10, γ 10 are constants, for i = 1,, n We can then show that the two-soliton solution, as given by 1, is equivalent to the Wronskian θ Wθ 1, θ = 1 θ θ 1,x θ,x = q q 1 e γ 1+γ 1 + q p 1 q q 1 e Λ 1 + p q 1 q q 1 e Λ + p p 1 q q 1 e Λ 1+Λ 16, 17

21 CHAPTER THE KP AND MKP EQUATIONS 0 in which Λ i = η i γ i + Λ i0, i = 1, and Λ i0 invariant under the transformation τ q q 1 e γ 1+γ τ, is the phase-constant Since u = log τ xx is Wθ 1, θ 1 + q p 1 q q 1 e Λ 1 + p q 1 q q 1 e Λ + p p 1 q q 1 e Λ 1+Λ By choosing Λ 10 = log q p 1 q q 1 and Λ 0 = log p q 1 q q 1, the Wronskian Wθ 1, θ may be written as 1 + e Λ 1 + e Λ + p 1 p q 1 q p 1 q q 1 p eλ 1+Λ, which is equal to 15 This result has been generalised [1, 45] to express the n-soliton solution of the KP equation as the n n Wronskian τ = Wθ 1, θ,, θ n, 18 where θ 1 θ n θ 1 Wθ 1, θ,, θ n = 1 θ 1 n θ n 1 1 θ n n 1 19 and θ k := k θ x k By choosing p n > q n > > p 1 > q 1, the Wronskian 19 is positive-definite and the n-soliton solution is regular Solutions of the KdV equation can be recovered by setting p i = q i = λ i For example, the one-soliton Wronskian solution of the KP equation reduces from u = 1 1 p 1 q 1 sech Λ 1 to u = λ 1 sech λ 1 x 4λ 1t 13 Wronskian solutions obtained from Darboux transformations In this section, we introduce an alternative method to Hirota s for finding multi-soliton solutions of a nonlinear evolution equation As far back as 188, the French mathematician Jean Gaston Darboux proved [13] that the Sturm-Louville equation d y + λ υx y = 0, 0 dx

22 CHAPTER THE KP AND MKP EQUATIONS 1 with ψ as a fixed solution, is covariant with respect to the transformation y ỹ = dy dx ψ xψ 1 y and υ υ = υ logψ x Darboux s result means that ỹ satisfies the Sturm-Louville equation 0 with potential υ, so that d ỹ + λ υx ỹ = 0 dx Almost a century later, in 1979, Matveev [37] realised that a similar covariance property as Darboux s for the Sturm-Louville equation holds for all equations of the form n m f f t = υ m x m, 1 where f = fx, t and υ = υx, t m=0 Returning to the KP equation, let θ = θx, y, t be an eigenfunction for L KP and M KP so that L KP [θ] = M KP [θ] = 0, which imply θ xx + v x θ θ y = 0, 4θ xxx + 6v x θ x + 3v xx θ + 3v xx θ + 3v y + θ t = 0 3 Equations and 3 are of similar form to those proposed by Matveev and are therefore Darboux covariant To generate a new solution, we consider another pair of operators L KP = G θ L KP G 1 θ operator By observing that and M KP = G θ M KP G 1 θ in which G θ is an invertible differential [ L KP, M KP ] = GL KP M KP G 1 GM KP L KP G 1 = G[L KP, M KP ]G 1 = 0, we can see that L KP, M KP are compatible if and only if L KP, M KP are compatible Definition 1 A Darboux transformation from L KP, M KP to L KP, M KP is defined by G θ = θ x θ 1 such that G θ [0] = 0 Let φ be another eigenfunction for L KP, M KP Then L KP [G θ [φ]] = G θ L KP G 1 θ [G θ[φ]] = G θ [L KP [φ]] = G θ [0] = 0 and similarly, MKP [G θ [φ]] = 0 Therefore, φ := G θ [φ] is an eigenfunction for L KP, MKP Calculating L KP gives L KP = x + u + θ xx θ 1 θ xθ y = x + u + log θ xx y = x + ũ y

23 CHAPTER THE KP AND MKP EQUATIONS So the effect of the Darboux transformation is that u ũ = u + log θ xx 4 We may conclude that if u is a solution of the KP equation and if L[θ] = 0 = M[θ], then ũ satisfies the KP equation too, that is: ũ t + 6ũũ x + ũ xxx x + 3ũ yy = 0 Repeating the process of determining ũ using M KP gives entirely consistent results If we take the trivial vacuum solution u = 0, from L[θ] = 0 = M[θ] we obtain θ y = θ xx and θ t = 4θ xxx 5 We choose the simplest solution of equations 5, which is θ = e η 1 + e γ 1 Here, we are using the same notation γ i, η i and Λ i as in the previous section substitution of this choice of θ into 4, we have the one-soliton solution ũ = 1 1 sech Λ 1 Upon In 1955, Crum [4] considered iterating Darboux s result and showed that the iterated solution could be formulated as the Wronskian determinant of eigenfunctions He also showed that the Darboux transformation adds an eigenvalue to the spectrum of the Schrödinger operator: for nonlinear evolution equations such as the KP and mkp equations, this means that a soliton is added by each Darboux transformation The key to this iteration is the transformation of the eigenfunction φ Let θ i, i = 1,,, n be a particular set of invertible, distinct eigenfunctions Furthermore, let φ = φ [1] be an eigenfunction for L KP[1] = L KP and θ [1] = θ 1 So φ [] = G θ[1] [φ [1] ] = φ x θ 1,x θ 1 / θ = 1 φ θ 1, θ 1,x φ x 1 φ is an eigenfunction for L KP[] For the second iteration θ 1 θ φ / θ 1 θ φ [3] = G θ[] [φ [] ] = θ 1,x θ,x φ x θ 1,x θ,x, θ 1,xx θ,xx φ xx

24 CHAPTER THE KP AND MKP EQUATIONS 3 in which θ [] = φ [] φ θ, is an eigenfunction for L KP[3] Here, we have the linear equations θ i,y = θ i,xx and θ i,t = 4θ i,xxx, for i = 1,,, n and we choose the solutions θ i = e η i + e γ i After n iterations, for n 1, we have φ [n+1] = Wθ 1, θ,, θ n, φ Wθ 1, θ,, θ n, 6 where θ [k] = φ [k] φ θk From each G θ[k] we obtain a new compatible Lax pair L KP[n+1], M KP[n+1], from which we obtain a new solution u [n+1] This class of solutions can be written compactly using the Wronskian determinant For n 1, we have: u [n+1] = u + log Wθ 1, θ,, θ n xx 7 We can also obtain this family of solutions by transforming the pseudo-differential operator L KP In [43] the authors show how to obtain the one-soliton solution from the Darboux transformation L KP = G θ L KP G 1 θ To see why the Darboux transformation L KP = G θ L KP G 1 θ lemma [43]: works, we need the following Lemma 1 Let L = L KP If L = G θ LG 1 θ and φ = G θ [φ], then L xq [P 0 L q, L] G θ L xq [P 0 L q, L]G 1 θ [θθ 1 [θ xq P 0 L q θ] x 1 θ 1, L], [ φ xq P 0 L q φ] φθ 1 [θ xq P 0 L q θ] x + θθ 1 [φ xq P 0 L q φ] x The notation [b] is used to denote multiplication with the function b The eigenfunction for the hierarchy L xq = [P 0 L q, L], where L = L KP, is the function θ = θx, x q satisfying the linear equations θ xq = P 0 L q θ, q = 1,, 3, The above equations are compatible and may be considered simultaneously for different q s With this definition of the eigenfunction θ, we may deduce from Lemma 1 that L xq [P 0 L q, L] 0 and [ φ xq P 0 L q φ] 0

25 CHAPTER THE KP AND MKP EQUATIONS 4 So if L satisfies the KP hierarchy with eigenfunctions θ and φ, then L xq = G θ LG 1 θ satisfies the hierarchy L xq = [P 0 L q, L] Furthermore, φ = G θ [φ] is an eigenfunction for L xq = [P 0 L q, L], so that φ satisfies the linear equations Let us now calculate L We find that L = θ x θ 1 x + 1 u 1 x φ xq = P 0 L q φ, q = 1,, 3, + u x = x + 1 u + lnθ xx 1 x + + x + So the effect of the Darboux transformation is that u ũ = u + lnθ xx, + u 3 3 θ x 1 θ 1 u + 1 u x uθ 1,x θx 1 θ x θ xx θ + θxθ 3 3 x u ũ = u + 1 u x uθ x θ 1 θ x θ xx θ + θ 3 xθ 3, The coefficients of L will satisfy 9 and 10 equation 1 In particular, ũ will satisfy the KP By considering n distinct eigenfunctions θ i, i = 1,, n, the Darboux transformation can be iterated so that, schematically, L KP[1] G θ[1] L KP[] G θ[] Gθ [n] L KP[n+1], for n 1 By taking the vacuum solution u = 0, we again obtain 6 and 7 14 Grammian solutions obtained from Hirota s method Grammians are determinants of matrices whose entries are in integral form For the KP equation 1, the solution τ can be written as the Grammian detg [9], where c 1,1 + x θ 1 ρ 1 dx c i,n + x θ 1 ρ n dx G = c n,1 + x θ nρ 1 dx c n,n +, x θ n ρ n dx and, for i, j = 1,,, n, the c i,j are constants Both θ i and ρ i, i = 1,,, n are functions of x, y and t satisfying the linear equations θ i,y = θ i,xx, θ i,t = 4θ i,xxx, ρ i,xx = ρ i,y, ρ i,t = 4ρ i,xxx

26 CHAPTER THE KP AND MKP EQUATIONS 5 By choosing the solutions of these equations to be θ = e η i and ρ = e γ i, with η i = p i x + p i y 4p i t and γ i = q i x + q i y 4q i t, the one-soliton solution is u = logτ xx = 1 p q sech 1 Λ + ξ, 8 in which τ = p q eη γ, Λ = η γ and ξ = log 1 p q is the phase-constant For the n-soliton solution, u = logτ xx, 9 where τ = det G and the entries in G are of the form G i,j = δ i,j + 1 p i q j e η i γ j By choosing p n > q n > p 1 > q 1, detg will be positive-definite and the n-soliton solution will be regular By setting p i = q i = λ i, we recover solutions of the KdV equation For example, the one-soliton solution u = 1 p q sech 1 Λ + ξ reduces to u = λ sech λ x 4λ t λ log λ 15 Grammian solutions obtained from binary Darboux transformations Grammian solutions of the KP equation can also be found using binary Darboux transformations where there are two eigenfunctions transforming The additional eigenfunction comes from an adjoint system Let H be a Hilbert space with an inner product,, and let A : H H be a differential operator Then there exists a differential operator A : H H with the property A[a], b = a, A[b] for all a, b H This gives us the properties 1 AB = B A, A = A, 3 A + B = A + B,

27 CHAPTER THE KP AND MKP EQUATIONS 6 4 If A is invertible, so is A and then A 1 = A 1 = A For matrix differential operators acting on complex vectors, a, b = + b a dx Then integration by parts gives us u x i = 1 i xu i, which we can use for all partial derivatives in differential operators Binary Darboux transformations for the KP hierarchy have been discussed in [44] However, here we shall only give the construction in terms of the Lax pair We now calculate the adjoint of the Lax pair L KP, M KP, which is L KP = x + v x + y, 30 M KP = 4 3 x 6v x x 3v xx + 3v y t 31 If [L KP, M KP ] = 0, then [L KP, M KP] = 0 and the compatibility condition [L KP, M KP] = 0 gives u t + u xxx + 6u u x x + 3u yy = 0, which is the adjoint of 1, the KP equation G θ We have seen that L KP = G θ L KP G 1 θ with G θ = θ x θ 1 The adjoint of L KP is L KP = G θ L KPG θ which can be rearranged to give L KP = G θ L KPG θ Similarly M KP = M KPG θ So the Darboux transformation from L KP, M KP to L KP, M KP induces an adjoint Darboux transformation in the opposite direction from L KP, M KP to L KP, M KP To describe the general form of the binary Darboux transformation, we consider another Lax pair ˆL KP, ˆM KP with eigenfunction ˆθ such that Gˆθ : ˆL KP, ˆM KP L KP, M KP Then we have the mapping G 1 ˆθ G θ : L KP, M KP ˆL KP, ˆM KP However, this mapping can only be defined if we can determine ˆθ This can be achieved by first noticing that, from ker G θ we obtain some nontrivial solution of the equations L KP[θ] = M KP[θ] = 0, which we denote by iθ The equation G θ [iθ] = 0 is satisfied by iθ = θ Now, corresponding to ˆθ ker ˆL KP ker ˆM KP, there exists a solution iˆθ ker L KP ker M KP We can then use the mapping G θ : L KP, M KP L KP, M KP to obtain ˆθ = i 1 G θ [ρ] for any ρ ker L KP ker M KP This enables us to define the binary Darboux transformation for the KP equation

28 CHAPTER THE KP AND MKP EQUATIONS 7 G θ,ρ Gˆθ L, M G θ L, M ˆL, ˆM θ,φ ˆθ L, M G L, M G ˆθ θ ˆL, ˆM ψ,ρ iˆθ Figure 1: Construction of the binary Darboux transformation Definition For ρ ker L KP ker M KP, we define G θ,ρ = G 1 ˆθ G ρ, where ˆθ = i 1 G θ [ρ] A binary Darboux transformation from L KP, M KP to ˆL KP, ˆM KP is defined by G θ,ρ such that G θ,ρ [0] = 0 Figure 1 illustrates the construction of the binary Darboux transformation To determine the binary Darboux transformation we must calculate ˆθ We have ˆθ = G θ [ρ] = θ x 1 θ ρ = θω 1, where So the binary Darboux transformation is A similar calculation gives us G θ,ρ = G 1 ˆθ Ω = 1 x [ρ θ] 3 G ρ = θω 1 1 Ω x θ 1 = θω 1 x 1 Ω Ω x θ 1 = 1 θω 1 1 x ρ G θ,ρ = 1 ρω 1 x θ Let ψ be another eigenfunction for L KP, M KP Then ˆL KP[G θ,ρ [ψ]] = G θ,ρ L KPG θ,ρ [G θ,ρ x [ψ]] = G θ,ρ L KP[ψ] = G θ,ρ [0] = 0,

29 CHAPTER THE KP AND MKP EQUATIONS 8 and similarly, M KP[G θ,ρ [ψ]] = 0 Therefore, ˆψ := G θ,ρ [ψ] is an eigenfunction for ˆL KP, ˆM KP To calculate û, we use the fact that both L KP, M KP and ˆL KP, ˆM KP map to L KP, M KP So we have that Then isolating û gives u + logθ xx = û + logθω 1 θ, ρ xx û = u + logωθ, ρ xx As was the case with the Darboux transformations, the binary Darboux transformation can be iterated to give an infinite family of solutions of the KP equation a particular set of invertible, distinct eigenfunctions of L KP[i+1] particular set of invertible, distinct eigenfunctions for L KP[i+1] Let θ i be and let ρ i and ψ i be a for i = 1,,, n Then the formulae for the nth binary Darboux transformations for the eigenfunction φ and the binary eigenfunction ψ are: / ΩΘ, P Ωφ, P ΩΘ, φ [n+1] = P, Θ φ ΩΘ, P ψ [n+1] = ΩΘ, ψ / ΩΘ, P P ψ and Ωφ [n+1], ψ [n+1] = / ΩΘ, P Ωφ, P ΩΘ, P ΩΘ, ψ Ωφ, ψ In the above formulae, Θ = θ 1, θ,, θ n, P = ρ 1, ρ,, ρ n and Ω is defined by 3 The class of solutions u [n+1] can be written compactly using the Grammian For n 1, u [n+1] = u + log ΩΘ, P 33 xx For soliton solutions, we take the trivial vacuum solution u = 0 Then the eigenfunctions θ i and the binary eigenfunctions ρ i satisfy θ i,y = θ i,xx, θ i,t = 4θ i,xxx, ρ i,xx = ρ i,y, ρ i,t = 4ρ i,xxx Finally, we choose the solutions of the above to equations to be θ = e η i and ρ = e γ i, with η i = p i x + p i y 4p i t and γ i = q i x + q i y 4q i t We then obtain the one-soliton solution 8

30 CHAPTER THE KP AND MKP EQUATIONS 9 The modified Kadomstsev-Petviashvili equation The mkp equation is w xt + w xxx 6w w x x + 3w yy + 6w x w y + 6w xx w y dx = 0, 34 which can also be written in potential form V tx + V xxxx 6VxV xx + 3V yy + 6V xx V y = 0, 35 where w = V x It originated with Dubrovsky and Konopelchenko in [3] By neglecting the y-derivative term in 34, we recover the mkdv equation 118 The mkp equation has the Lax pair L mkp = x + V x x y, M mkp = 4 x 3 + 1V x x + 6V xx + Vx V y x + t 1 The mkp hierarchy We now construct the mkp hierarchy using a similar analysis to that of the KP hierarchy Here, we use the pseudodifferential operator L mkp = x + w + w 1 1 x Let L = L mkp The mkp hierarchy is defined to be + w x + 36 L xq = [P 1 L q, L], q = 1,, 3, 37 The first four projections are the differential operators: P 1 L = x, P 1 L = x + w x, 38 P 1 L 3 = 3 x + 3w x + 3w x + w + w 1 x, P 1 L 4 = 4 x + 4w 3 x + 6w x + 4w 1 + 6w x + 4w 3 + 6w 1x + 4w xx + 4w + 1ww x + 1ww 1 x The evolution equation 37 gives w x1 = w x, w 1x1 = w 1x, L x1 = [P 1 L, L] w x1 = w x,, 39

31 CHAPTER THE KP AND MKP EQUATIONS 30 L x = [P 1 L, L] L x3 = [P 1 L 3, L] w y = w xx + ww x + w 1x, w 1y = w 1xx + ww 1 x + w x, w y = w xx w 1 w xx + ww x + 4w x w + w 3x,, w t = w xxx + 3w 1xx + 3w x + 3ww x x + 3w w x +6ww 1 x,, L x4 = [P 1 L 4, L] w x4 = w xxxx + 6w xx + 4w 1xxx + 4w 3x + 4ww xxx +10w x w xx + 6w w xx + 1wwx + 4w 3 w x +1ww 1xx + 18w x w 1x + 1w 1 w 1x +6w 1 w xx + 1w w 1x + 4ww x w 1 + 1ww x +1w x w,, 4 where again we have set x 1 = x, x = y, x 3 = t These are the first four equations of the mkp hierarchy We can eliminate the fields w 1, w, by expressing them in terms of the field w and its x- and y-derivatives Eliminating w 1 and w via 40 allows us to rewrite the first component in 41 in terms of w and its x- and y-derivatives From this we obtain the mkp equation 34, where the scaling t 4t has been made Wronskian solutions obtained from Hirota s method The dependent variable transformation [30] for the mkp equation is τ V = log τ Substituting this into the mkp equation 34 gives the bilinear form [9, 30] D x + D y τ τ = 0, 43 D 3 x + D t 3D x D y τ τ = 0 44 The n-soliton solution of the mkp equation [9] can be expressed as w = 1 n log, τ τ τ = Wθ 1, θ,, θ n, τ = Wθ 1,x, θ,x,, θ n,x,

32 CHAPTER THE KP AND MKP EQUATIONS 31 in which θ i = e η i +e γ i, η i = p i x+p i y 4p i t and γ i = q i x+q i y 4qi t for i = 1,,, n Both u = log τ xx and ú = log τ xx are solutions of the KP equation 1 The one-soliton Wronskian solution of the mkp equation is θx w = log θ x = 1 4 p 1 q 1 p 1 q sech Λ 1 sech Λ 1 + log where Λ 1 = η 1 γ 1 p1 q 1 45, 46 3 Wronskian solutions obtained from Darboux transformations Darboux transformations can also be used to find a family of Wronskian solutions of the mkp equation To obtain Wronskian solutions, we use a different differential operator G θ from the KP equation For the mkp equation, each of L mkp, L mkp eigenfunction θ, is covariant under the Darboux transformation [41, 43] and M mkp, with G θ = θ 1 x 1 x θ 1 = 1 θθ x 1 x We focus on L mkp here, using the following lemma [43]: Lemma Let L = L mkp 1 If L = θ 1 Lθ and φ = θ 1 φ then L xq [P 1 L q, L] θ 1 L xq [P 1 L q, L]θ [θ 1 [θ xq P 1 L q θ], L], [ φ xq P 1 L q φ] θ φ [θ xq P 1 L q θ] + θ 1 [φ xq P 1 L q ψ] 47 If L = θ 1 x x L 1 x L xq [P 1 L q, L] θ 1 x θ x and φ = θ 1 φ x then x L xq [P 1 L q, L] 1 θ x [θ 1 [θ xq P 1 L q θ] x, L], [ φ xq P 1 L q φ] θ xφ x [θ xq P 1 L q θ] x + θ 1 x [φ xq P 1 L q φ] x 48 The eigenfunction for the hierarchy L xq = [P 1 L q, L], where L = L mkp, is the function θ = θx, x q satisfying the linear equations θ xq = P 1 L q θ, q = 1,, 3, The above equations are compatible and may be considered simultaneously for different q s With this definition of the eigenfunction θ, we may deduce from Lemma that L xq [P 1 L q, L] 0 and [ φ xq P 1 L q φ] 0 x x

33 CHAPTER THE KP AND MKP EQUATIONS 3 So if L satisfies the mkp hierarchy with eigenfunctions θ and φ, then L xq and L xq = θlθ 1 = θx 1 x L x 1 θ x satisfy the hierarchy L xq = [P 1 L q, L] However, it is the composition of the two aforementioned transformations that we are interested in here since this will give us Wronskian solutions In addition to θ and φ, the constant 1 is a trivial eigenfunction Then from part 1 of Lemma, Lxq = θlθ 1 with eigenfunctions φ = θ 1 φ and θ 1 Using the eigenfunction θ 1 in part of Lemma gives L L = θ 1 x 1 1 x L x θ 1 1 x = G θ LG θ, with eigenfunction φ = φ θθx 1 φ x So if L satisfies the mkp hierarchy with eigenfunctions θ and φ, then L xq = G θ LG 1 θ satisfies the hierarchy L xq = [P 1 L q, L] Furthermore, φ = G θ [φ] is an eigenfunction for L xq equations = [P 1 L q, L], so that φ satisfies the linear φ xq = P 1 L q φ, q = 1,, 3, Upon calculation of L mkp = G θ LG 1 θ, we obtain [43] L mkp = x + w + θx 1 θ xx θ 1 θ x + w 1 + w x + θ 1 θ x x x 1 + So the effect of the Darboux transformation is that θx w w = w + log, θ x w 1 w 1 = w 1 + w x + θ 1 θ x x, The coefficients of L will satisfy 40, 41 and 4 In particular, w will satisfy the mkp equation 34 Let θ i, i = 1,,, n be a particular set of invertible, distinct eigenfunctions Furthermore, let φ = φ [1] be an eigenfunction for L mkp[1] = L mkp and θ [1] = θ 1 Then φ [] = φ θx 1 φ x = θ 1 θ 1,x / φ φ x θ x

34 CHAPTER THE KP AND MKP EQUATIONS 33 is an eigenfunction for L mkp[] For the second transformation, θ φ [3] = φ 1,x θ,x θ 1,xx θ,xx φ θ 1 θ x θ 1,xx θ,xx + φ θ 1 θ 1,x xx / θ 1,x θ,x θ θ,x θ 1,xx θ,xx θ 1 θ φ / θ 1,x θ,x = θ 1,x θ,x φ x θ 1,xx θ,xx θ 1,xx θ,xx φ xx = Wθ 1, θ, φ Wθ 1,x, θ,x, in which θ [] = φ [] φ θ, is an eigenfunction for L mkp[3] In general, for n 1 φ [n+1] = 1 n Wθ 1, θ,, θ n, φ Wθ 1,x, θ,x,, θ n,x, where θ [k] = φ [k] φ θk From each G θ[k] we obtain a new covariant L mkp[n+1] from which we obtain a new solution w [n+1] This class of solutions can be written compactly using the Wronskian determinant For n 1, we have log w [n+1] = w + 1 n Wθ1, θ,, θ n 49 log Wθ 1,x, θ,x,, θ n,x x 4 Grammian solutions obtained from Hirota s method For the mkp equation 35, both τ n and τ n+1 can be written as the Grammian determinants detg and detǵ [9], where the entries of G are of the form G i,j = δ i,j p j q i p i q j eη i γ j and the entries of Ǵ are of the form Ǵ i,j = δ i,j e η i γ j p i q j for i, j = 1,,, n For example, the one-soliton solution is w = 1 4 pq 1 Λ + ϕ p q sech sech where ϕ = log p qp q and χ = log 1 p q For the n-soliton solution, w = Λ + χ, 50 τ log, 51 τ xx

35 CHAPTER THE KP AND MKP EQUATIONS 34 where τ = detg and τ = detǵ Both u = logτ xx and ú = log τ xx are solutions of the KP equation By choosing p n > q n > p 1 > q 1 > 0 or 0 > p n > q n > p 1 > q 1, both detg and detǵ will be positive-definite and the n-soliton solution will be regular 5 Grammian solutions obtained from Darboux transformations Grammian solutions of the mkp equation can also be found from binary Darboux transformations We use the same construction as for the KP equation as illustrated in Figure 1 As a notational convenience, we denote an element of L mkp M mkp as ρ x rather than ρ The equation G θ [iθ] is satisfied by iθ = θ x To determine ˆθ, we need the integrals Ω = x 1 [ρ θ x ] and Ω = x 1 [ρ xθ], where Ω + Ω = ρ θ We have that and therefore iˆθ = ˆθ x = G θ [ρ x] = θ x x 1 θ ρ x ˆθ 1 x = x 1 ρ xθθ 1 x = Ω θ 1 x We can then isolate ˆθ by integrating by parts and then taking inverses Doing so gives ˆθ = Ω θ + x 1 Ω xθ 1 1 = ρ Ω θ 1 1 = θω 1 Now that we have determined ˆθ, we may obtain G θ,ρx = ˆθ 1 x ˆθ 1 x θ 1 1 x x θ 1 = θω 1 x 1 Ω x θ 1 = 1 θω 1 1 x ρ x

36 CHAPTER THE KP AND MKP EQUATIONS 35 A similar calculation gives G θ,ρ x = 1 x ρω 1 x θ Let ψ be another eigenfunction for L mkp, M mkp Then ˆL mkp[g θ,ρ [ψ]] = G θ,ρ L mkpg θ,ρ [G θ,ρ [ψ]] = G θ,ρ L mkp[ψ] = G θ,ρ [0] = 0, and similarly, M mkp[g θ,ρ [ψ]] = 0 Therefore, ˆψ := G θ,ρ [ψ] is an eigenfunction for ˆL mkp, ˆM mkp To calculate ŵ, we use the fact that both L mkp, M mkp and ˆL mkp, ˆM mkp map to L mkp, M mkp So we have that θx ˆθx w + log = ŵ + log θ x ˆθ Then isolating ŵ gives the one-soliton solution ŵ = w + log1 θρ Ω 1 θ, ρ x Ωθ, ρ ρ = w + log / Ωθ, ρ θ 1 The binary Darboux transformation can be iterated to give an infinite family of solutions of the mkp equation Let θ i be a particular set of invertible, distinct eigenfunctions of L mkp[i+1] and let ρ i and ψ i be a particular set of invertible, distinct eigenfunctions for L mkp[i+1] for i = 1,,, n Furthermore, let Θ = θ 1, θ,, θ n and P = ρ 1, ρ,, ρ n Then the formulae for the nth binary Darboux transformations for the eigenfunction φ and the binary eigenfunction ψ are: / ΩΘ, P Ωφ, P ΩΘ, φ [n+1] = P, Θ φ ΩΘ, P ψ [n+1] = ΩΘ, ψ / ΩΘ, P P ψ x x and Ωφ [n+1], ψ [n+1] = / ΩΘ, P Ωφ, P ΩΘ, P ΩΘ, ψ Ωφ, ψ The class of solutions w [n+1] can be written compactly using the Grammian For n 1, ΩΘ, P P w [n+1] = w + log / ΩΘ, P 5 Θ I x

37 CHAPTER THE KP AND MKP EQUATIONS 36 For soliton solutions, we take the trivial vacuum solution w = 0 Then the eigenfunctions θ i and the binary eigenfunctions ρ i satisfy θ i,y = θ i,xx, θ i,t = 4θ i,xxx, ρ i,xx = ρ i,y, ρ i,t = 4ρ i,xxx Finally, we choose the solutions of the above equations to be θ = e η i and ρ = e γ i, with η i = p i x + p i y 4p i t and γ i = q i x + q i y 4q i t We then obtain the one-soliton solution 50 3 Direct verification of the solutions 31 Derivatives of Wronskian determinants Consider the n-vector Θ = θ 1, θ,, θ n T depending on x 1 = x and possibly other variables x, x 3, Using the notation Θ i to denote the ith x-derivative of Θ with respect to x, we can define the Wronskian determinant τ = Θ 0, Θ 1,, Θ n 1 j 1 θ i = det x j 1, 1 i, j n The derivatives of τ with respect to x j can be calculated from the basic result n 1 τ xj = Θ 0,, Θ i 1, Θ i x j, Θ i+1,, Θ n 1 i=0 For example, since a determinant with two equal rows or columns vanishes, Differentiating with respect to x again gives τ x = Θ 0, Θ 1,, Θ n, Θ n τ xx = Θ 0, Θ 1,, Θ n, Θ n+1 + Θ 0, Θ 1,, Θ n 3, Θ n 1, Θ n We can use a partition notation to denote derivatives of the Wronskian determinant τ If we take a partition λ = λ 1, λ,, λ p, a sequence of positive integers, where λ 1 λ λ p, then we can write derivatives of τ as For example, τ λ := τ xλ1 x λ x λp 53 τ x x 1 = τ 1 and τ x1 x 1 = τ 1

38 CHAPTER THE KP AND MKP EQUATIONS 37 For the Wronskian determinant 19 this notation can be used to denote shifts in the index of the columns For example, and in general, W λ := W 1 := Θ 0 Θ 1 Θ n Θ n W := Θ 0 Θ 1 Θ n Θ n+1, Θ 0 Θ n p 1 Θ n p+λp Θ n 1+λ 1 If Θ satisfies the linear equations Θ xj = Θ j, j = 1,, 3, 54 then we can write down the relationship between the derivatives τ λ and the determinants W λ ; τ λ = µ ζ µ λ W µ, 55 where the sum is over all partitions µ and the matrices ζ µ λ symmetric group S n Some useful derivatives of τ are [ ] [ ] τ 1 = [1] W 1, τ = 1 1 W, τ W 1 τ τ 1 = τ τ τ τ = τ τ 1 4 W 3 W 1 W 1 3 The relations 56 can be inverted using the formula are the character tables for the, 56 W 4 W 31 W W 1 W 1 4 W = λ l 1 λ ζµ λ τ λ,

39 CHAPTER THE KP AND MKP EQUATIONS 38 where for a partition λ = r mr,, m, 1 m 1, l λ := r i=1 im i m i! Thus [ ] [ ] W 1 = [1] τ 1, W = τ, W τ 1 W τ W 1 = τ 6 1, 57 W τ 1 3 W τ 4 W τ W = τ 4 W τ W Laplace expansion of determinants A Laplace expansion is an expression of an nth-order determinant as a sum of products of rth- and n rth-order determinants Consider an n n matrix A with determinant deta τ 1 4 We use Υ j 1,,j m 1,,m to denote the m m determinant taken from deta, where 1,, m and j 1,, j m m = n/ are the rows and columns respectively of deta Furthermore, Ξ j 1,,j m 1,,m denotes the m m determinant obtained by deleting the 1,, m rows and j 1,, j m columns of deta We can now define A l, the Laplace expansion of deta, as A l = j 1 <j <<j m m+j1+ +jm Υ j1,,jm 1,,,m Ξj 1,,j m 1,,,m 58 For example, if n = 4, then deta = a 11 a 1 a 13 a 14 a 1 a a 3 a 4 a 31 a 3 a 33 a 34 a 41 a 4 a 43 a 44,

40 CHAPTER THE KP AND MKP EQUATIONS 39 Υ 1, 1, = a 11 a 1 a 1 a, Ξ1, 1, = a 33 a 34 a 43 a 44 etc, and a A l = 11 a 1 a 33 a 34 a 1 a a 43 a 44 a 11 a 13 a 3 a 34 a 1 a 3 a 4 a 44 + a 11 a 14 a 3 a 33 a 1 a 4 a 4 a 43 a + 1 a 13 a 31 a 34 a a 3 a 41 a 44 a 1 a 14 a 31 a 33 a a 4 a 41 a 43 + a 13 a 14 a 31 a 3 a 3 a 4 a 41 a 4 33 Plücker relations The term Plücker relation will be used to describe a type of quadratic identity amongst determinants W µ Consider the n n determinant = Θ0 Θ n , Θ 0 Θ n 3 Θ n Θ n 1 Θ n Θ n+1 where 0 denotes the zero n-vector Applying 58, the Laplace expansion of 59 results in = 0 This is the simplest case of a Plücker relation We can now rearrange the right-hand side of 59 by adding the n + kth row to the kth row k = 1,,, n and subtracting the kth column from the n kth column k = 1,,, n This gives = Θ0 Θ n Θ n Θ n 1 Θ n Θ n Θ 0 Θ n 3 Θ n Θ n 1 Θ n Θ n+1 From the Laplace expansion of 60 we obtain the identity Θ 0 Θ n 3 Θ n Θ n+1 Θ 0 Θ n 1 Θ 0 Θ n 3 Θ n 1 Θ n+1 Θ 0 Θ n Θ n + Θ 0 Θ n 3 Θ n 1 Θ n Θ 0 Θ n Θ n+1 = 0, which in our partition notation is W W W 1 W 1 + W W 1 = 0 61 To verify the Wronksian solution of the KP equation, we need to show that the left hand side of 13 with t 4t, that is τ τ τ 31 τ 4τ 1 3 τ 3 τ 1 + 3τ 1 τ,

41 CHAPTER THE KP AND MKP EQUATIONS 40 vanishes when τ = W Using the tables 56, this condition becomes W W W 1 W 1 + W W 1 = 0, which is the Plücker relation 61 Hence the solution is verified For the mkp equation, the left hand sides of the coupled system can be written in partition notation as τ 1 + τ τ τ 1 τ 1 + τ 1 τ τ, 6 τ τ 3 3 τ 1 τ + 3τ 1 τ 1 4 τ 3 τ 3 τ 1 τ τ 1 τ τ τ τ 1 τ, 63 which are again Plücker relations see for example [9] and identically zero Hence the solution is verified 34 Derivatives of Grammian determinants In addition to the vector Θ previously introduced, consider another vector P = ρ 1, ρ,, ρ n T also depending on x 1, x, x 3, For any n n matrix A whose entries a ij satisfy x a ij = α i β j, the derivative of its determinant can be written as x deta = n 1 i+j α i β j A i j i,j=1 α 1 A =, α n β 1 β n 0 where A i j is the i, jth minor of A So for the Grammian τ, its derivative with respect to x is the bordered determinant G Θ τ x = P T 0 Furthermore, if we assume that Θ and P satisfy Θ xj = Θ j and P xk = 1 k P k, j, k = 1,, 3,, then it follows that G ij k 1 = 1 m k 1 m θ i m ρ j x k x k 1 m x m m=0

42 CHAPTER THE KP AND MKP EQUATIONS 41 For derivatives of Grammians, we must use a Frobenius notation for partitions Consider the Young diagram associated with a partition λ and let α i denote the number of boxes to the right of the diagonal in the ith row and β i the number of boxes below the diagonal in the ith column The two sets of nonnegative integers, α i and β i, determine the partition which in Frobenius notation we denote as λ = α 1,, α p β 1,, β p For example, the Young diagram for the partition λ = would be where we have used a to mark the diagonal entries So, in Frobenius notation, 43 1 is As with the Wronskian, we can use a notation to relate the derivatives of the Grammian τ and its original form For any partition λ = α 1 α p β 1 β p we define G λ = G α1 α p β 1 β p = 1 p G Θ α 1 Θ αp P β P βp 0 0 Then, for example, G 0 1 = G Θ 0 P 1 0 and G = G Θ 1 Θ 0 P P The relationship between the derivatives τ λ and G µ is the same as in 55 but with W replaced by G, ie τ λ = µ ζ µ λ G µ and the matrices ζ µ λ are as given in Jacobi identity for determinants The Grammian equivalent of a Laplace expansion is Jacobi s identity [1], which relates different size determinants Here we give the basic Jacobi identity Let A be an m m

43 CHAPTER THE KP AND MKP EQUATIONS 4 matrix We denote by A i,,j k,,l, the minor obtained by omitting the ith,, jth rows and the kth,, lth columns With this notation, the formula detaa i,j k,l = A i k A j k A i l A j l = A i k Aj l Aj k Ai l is a general case of the Jacobi identity In this identity, if deta is identified with, for example, G 10 10, with i, j as the last two rows and k, l as the last two columns then A i k = G 1 1, A i l = G 0 1, A j k = G 1 0, A j l = G 0 0, and A i,j k,l = G Thus Jacobi s identity gives G G G 1 1 G G 1 0 G 0 1 = 0 65 Rewriting this in partition notation gives G G G 1 G 1 + G G 1 = 0, 66 which is the same form as the simplest Plücker relation 61 for Wronskians To verify the Grammian solution, we need to show that the left hand side of 1 vanishes when τ = G Using equivalent tables to 56, we obtain G G G 1 1 G G 1 0 G 0 1 = 0, 67 which is the Plücker relation 65 Hence the solution is verified For the mkp equation, it can again be shown that see for example [9] equations 6 and 63 with τ = G and τ = Ǵ are both Plücker relations and identically zero, hence the solution is verified 4 The Miura transformation A Miura transformation between the KP and mkp heirarchies can be obtained from the following theorem [43]: Theorem 1 Let L satisfy the KP hierarchy 7 Then L = θ 1 Lθ satisfies the hierarchy L xq = [P 1 L q, L] with G[φ] = φ = θ 1 φ being an eigenfunction for L, that is φ xq = P 1 L q [φ]

44 CHAPTER THE KP AND MKP EQUATIONS 43 We shall now demonstrate this by implementing the gauge transformation L = θ 1 Lθ This gives L = + θ 1 θ x + 1 u 1 + u 1 uθ 1 θ x Comparing this with the operator and equating coefficients gives L mkp = x + w + w 1 1 x + w x + w 3 3 x + 69 w = θ 1 θ x, 70 w 1 = 1 u, 71 w = u 1 uθ 1 θ x 7 These coefficients will satisfy 40, 41 and 4 of the mkp hierarchy 37 Upon substitution of 71 into the first term of 41, we obtain u = V y V xx V x 73 Equation 73 is the Miura transformation between the KP and mkp equations Direct substitution of 73 into the KP equation 1 yields u t + u xxx + 6uu x x + 3u yy = y x V x x V tx + V xxxx 6V xv xx + 3V yy + 6V xx V y Therefore, if V x is a solution of the mkp equation 34, then the Miura transformation 73 defines a new solution of the KP equation 1

45 Chapter 3 A noncommutative KP equation In this chapter, we look at an example of a noncommuative nc integrable system Different approaches to this noncommutativity exist For example, the noncommutativity may arise through an underlying nc space defined by the noncommutativity of the coordinates: [x j, x k ] = iθ jk, where i = 1 and θ jk are real constants called the nc parameters For Euclidean spaces, the star-product is explicitly given by f gx := exp exp i θij x i fx gx x j x=x =x, 31 = fxgx + i θij x i fx x j gx + Oθ, 3 which is known as the Groenewold-Moyal product [4,40] The star-product is associative, that is f g h = f g h, and returns the ordinary product in the commutative limit θ jk 0 The star-product makes the ordinary spatial coordinate noncommutative, in that [x j, x k ] = x j x k x k x j = iθ jk In this case, an nckdv equation would have spacetime noncommutativity, that is [t, x] = iθ An nckp equation could have space-space noncommutativity in that [x, y] = iθ or space-time non-commutativity, in that either [t, x] = iθ or [t, y] = iθ Hamanaka and Toda [5 7] have extensively considered the cases where the Lax method and the Gelfand-Dickii hierarchies give nc equations defined on the nc space Lax [35], and later Goncharenko and Veselov [, 3], have considered a matrix version of the KdV equation and Gelfand and Etingof [10] have considered a quaternionic version of the KP equation 44

46 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 45 By dropping the assumption that the coefficients in the Lax pair or Gelfand-Dickii hierarchy commute, the results should be valid for all cases There are three levels of noncommutativity that could arise in the variables in an nc integrable system; the variables could be: Matrices with entries that commute, Matrices with entries that do not commute, for example a partitioned matrix, Noncommutative and not finite-dimensional matrices, for example, in [51], where the commutation relations [x j, x k ] = δ jk iθ jk are the Heisenberg algerba Where we have a noncommutative integrable system, we may still have associativity In the last chapter, we saw that multi-soliton solutions for the KP and mkp equations could be written as a logarithmic transformation, such as u = log τ xx, where τ could be the Wronskian or Grammian determinant We cannot use Hirota s bilinear method in the nc case as it does not make sense, for example, to take the derivative of the logarithm of a matrix This can be seen more clearly when attempting to differentiate the power series of loga for some function a = ax: loga = 1 a 1 a 1 a3 3 1 a If we were to differentiate 33 with respect to x we would obtain terms such as aa x + a x a aa x, so we have to discard making logarithmic transformations on dependent variables In addition, we cannot define the determinant of a matrix when its entries do not commute When this is the case, the natural replacement for a determinant is the quasideterminant 31 Quasideterminants The concept of quasideterminants originated with Gelfand and Retakh in [16] An n n matrix over a not necessarily commutative unital ring R has, in general, n quasideterminants We denote each quasideterminant by Z ij, 1 i, j n Let Z ij denote the matrix obtained from Z by deleting the ith row and jth column Let r j k be the row vector obtained from the kth row of Z by deleting the jth entry and let s i l be the column vector obtained from the lth row of Z by deleting the ith entry If Z ij is invertible, then Z ij

47 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 46 exists and Z ij = z ij r j i Zij 1 s i j 34 For example, if n = and Z = z ij, then there are 4 quasideterminants, one of which is z Z = 11 z 1 = z z 1 z11 1 z 1 z z 1 We shall henceforth adopt an alternative notation for quasideterminants by boxing the leading element For example, Z 1 = z 11 z 1 z 1 z = z 1 z 11 z 1 1 z Quasideterminants of Z can also be defined via the inverse of Z Suppose the matrix Z is invertible with inverse B = b ij If Z ij exists then Z ij = b 1 ji The theory of quasideterminants has been greatly developed over the years following their introduction, resulting in several properties and identities which were published by Gelfand, Gelfand, Retakh and Wilson in [14] Here, we recall some of the main results which we shall use in what follows 311 The matrix inverse The matrix inverse is given by z 1 z 11 z 1 11 z 1 z 1 z = z 1 z z 11 z 1 z 1 z provided that all inverses above exist n = 3 We have = 1 1 z 11 z 1 z 1 z z 11 z 1 z 1 z 1 1 z 11 z 1 z z 1 z z z 1 z 11 z1 1 1 z z 1 z , We can see this in the quasideterminant when z 11 z 1 z 13 Z = z 1 z z 3 z 31 z 3 z 33

48 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 47 and there are 9 quasideterminants For example: Z 13 = z 11 z 1 z 13 z 1 z z 3 z 31 z 3 z 33 = z 13 z 11 z 1 z 1 z z 31 z 3 1 z 3 z 33 = z 13 z 11 z 1 z 1 z z 1 3 z 31 1 z 31 z 3 z 1 z 1 1 z z 1 z 1 31 z 3 1 z 3 z 31 z 1 1 z 1 z 3 z 33 = z 13 z 11 z 1 z z 1 3 z 31 1 z 3 z 1 z z 1 z 1 31 z 3 1 z 3 z 11 z 31 z 3 z 1 z 1 1 z 33 z 1 z 3 z 31 z 1 1 z 1 z Noncommutative Jacobi identity Quasideterminants can be used to construct a noncommutative version of the Jacobi identity for determinants also called the noncommutative Sylvester s theorem in [14] One such case of this is given by A B C D f g E h i = A C E i A B E h A B D f 1 A C D g, 35 where A is a square matrix, B, C are column vectors, D, E are row vectors and f, g, h, i are single entries all of compatible length From these identities, we obtain the following row homology and column homology relations [14]: A B C D f g E h i A C D g 1 = A C E i A C D g 1 A B E h A B D f 1 36 = A B C D f g E h i A B D f 1 37

49 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 48 and A E B h 1 A B C D f g E h i = = A E A D B h B f 1 A E 1 C i A D A B C D f g E h i B f 1 A D C g Elementary row and column operations Replacing the expansion row by a left multiple has the effect of left multiplying the quasideterminant by that factor For example, E 0 A B EA EB = F g C d F A + gc F B + gd nn nn = g d CA 1 B A B = g 310 C d nn This is again true if we replace row with column operations and left- with right-multiplication All other row and column operations have no effect 314 Comparison with commutative determinants If Z is an n n matrix over a commutative ring R, then Z ij is related to det Z Z is invertible, then the j, ith entry of Z 1 = 1 i+j det Z i,j det Z noncommutative case, if Z 1 = B, then Z ij = b 1 ji So If Recall that in the c Z ij = 1 i+j det Z, 311 det Zi,j where the notation c = has been introduced to denote that the right-hand side of the equation is commutative For example, if n = then c Z 11 = 1 i+j det Z det Z i,j c = z 11z z 1 z 1 z c = z 11 z 1 z 1 z 1

50 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION Quasi-Plücker coordinates Given an n + k n matrix A, denote the ith row of A by A i, the submatrix of A having rows with indices in a subset I of {1,,, n + k} by A I Given i, j {1,,, n + k}, and I such that #I = n 1 and j I, the right quasi-plücker coordinates are defined by rij I = rija I A := I A I 1 A I 0 A i A j = A i 0, 31 ns ns A j 1 for any s {1, n} The second equality comes from Jacobi s identity and proves that the definition is independent of s Therefore we could also write the definition as rija I = A I A i The left quasi-plücker coordinates are defined in an analogous way For an n n + k A I A j 1 matrix B, lijb I = B I B j 1 B I B i = B I B i B j 0 0 1, in which we here denote the ith column of B by B i, and the submatrix of B having columns with indices in a subset I of {1,,, n + k} by B I The row homology and column homology relations can now be written in terms of quasi-plücker coordinates giving the following identities: A B C A B C A B C D f g = D f g D f g E h i E h i and A B C D f g E h i = A B 0 D f 0 E h 1 A B C D f g E h i A noncommutative KP hierarchy Gelfand-Dickii hierarchies in the nc setting have been considered in [7, 5, 7, 51]

51 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 50 In the nc case, with L = L KP, we have the differential operators P 0 L = x, P 0 L = x + u, 315 P 0 L 3 = 3 x + 3 u x + 3 u x + 3u, which, via the evolution equation 7, give the nckp hierarchy: L t1 = [P 0 L, L] u t1 = u x, u t1 = u x, u 3t1 = u 3x,, 316 L t = [P 0 L, L] u y = u xx + 4u x, u y = u xx + u 3x + 1 uu x + [u, u ], u 3y = u 3xx + u 4x 1 uu xx + u u x + [u, u 3 ],, 317 L t3 = [P 0 L 3, L] u t = u xxx + 6u xx + 6u 3x + 3 uu x + 3 u xu, u t = u xxx + 3u 3xx + 3u 4x + 3uu + 3 u xu + u u x + 3[u, u 3 ], 318 Eliminating u and u 3 via 317, and making the scaling t 4t allows us to rewrite the first component in 318 as v t + 3v x v x + v xxx x + 3v yy + 3[v x, v y ] = 0, 319 where again u = v x Equation 319 is the nckp equation nckp [46] in potential form When the variables in 319 do commute, we recover 1 Equation 319 could also be obtained from the Lax pair L KP = x + v x y, M KP = 4 3 x + 6v x x + 3v xx + 3v y + t, whose compatibility condition [L KP, M KP ] = 0 gives 319

52 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION Quasiwronskian solutions obtained from Darboux transformations Given that we can attempt to derive a noncommutative integrable system through the Lax method or the Gelfand-Dickii hierarchy, Darboux transformations appear to be the natural choice for finding multi-soliton solutions Both L KP and M KP are covariant under the Darboux transformation G θ = θ x θ 1 = x θ x θ 1 Let θ i, i = 1,, n, be a particular set of eigenfunctions It is assumed that, like the dependent variable u, the eigenfunction θ and its derivatives do not commute Introduce the notation Θ = θ 1,, θ n and ˆΘ = θ i 1 j i,j=1,,n, the n n Wronskian matrix of θ 1,, θ n Let φ = φ [1] be an eigenfunction of L KP[1] = L KP and θ [1] = θ 1 Then φ [] := G θ[1] [φ [1] ] and θ [] = φ [] φ θ are eigenfunctions for L KP[] = G θ[1] LG 1 θ [1] In general, for n 1 define the nth Darboux transform of φ by in which φ [n+1] = φ 1 [n] θ1 [n] θ 1 [n] φ [n], θ [k] = φ [k] φ θk It has been shown in [14] and in [3] that φ [n+1] can be expressed as Θ φ Θ n 1 φ n 1 Θ n φ n The effect of L KP = G θ L KP G 1 θ, MKP = G θ M KP G 1 θ is that ṽ = v + θ x θ 1

53 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 5 After n Darboux transformations we have v [n+1] = v + n i=1 θ [i],x θ 1 [i] Θ 0 = v Θ n 0 30 Θ n 1 1 Θ n 0 We call this type of quasideterminant in 30 a quasiwronskian 34 Quasigrammian solutions obtained from binary Darboux transformations A new family of solutions of nckp, obtained by binary Darboux transformations and expressible as quasideterminants, was introduced by Gilson and Nimmo in [19] adjoint Lax pair for nckp is The L KP = x + v x + y, M KP = 4 x 3 6v x x 3v xx + 3v y t Here, the notion of the adjoint has been extended from the well-known matrix case to any unital ring R, as considered by Matveev in [38]: suppose that for each a R, there exists a R, and for a derivative acting on R, x = x and for a product AB of elements of R, or operators on R, AB = B A Analogous to the commutative case, a potential Ωφ, ψ is introduced, satisfying Ωφ, ψ x = ψ φ, Ωφ, ψ y = ψ φ x ψ xφ, Ωφ, ψ t = 4ψ φ xx ψ xφ x +ψ xxφ 6ψ v x φ A binary Darboux transformation is defined by φ [n+1] = φ [n] θ [n] Ωθ [n], ρ [n] 1 Ωφ [n], ρ [n] and ψ [n+1] = ψ [n] ρ [n] Ωθ [n], ρ [n] Ωθ [n], ψ [n], in which θ [n] = φ [n] φ θn, ρ [n] = ψ [n] ψ ρn

54 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 53 Using the notation Θ = θ 1, θ n and P = ρ 1,, ρ n we have, for n 1 ΩΘ, P Ωφ, P φ [n+1] = Θ φ, ΩΘ, P ψ [n+1] = ΩΘ, ψ P ψ and ΩΘ, P Ωφ, P Ωφ [n+1], ψ [n+1] = ΩΘ, ψ Ωφ, ψ The effect of is that ˆL KP = G θ,φ L KP G 1 θ,φ, ˆMKP = G θ,φ M KP G 1 θ,φ ˆv = v + θωθ, ρ 1 ρ After n binary Darboux transformations we have n v [n+1] = v + θ [k] Ωθ [k], ρ [k] 1 ρ [k] k=1 ΩΘ, P P = v Θ 0 31 We call this type of quasideterminant in 31 a quasigrammian 35 Reduction to commutative Wronskian and Grammian solutions All of the quasideterminants expressing the Darboux-transformed eigenfunctions and potentials v [n+1] of nckp, should reduce to the corresponding commutative results in Chapter Using 311, in the commutative case, we have: The transformed eigenfunction Θ φ [n+1] = Θ n 1 Θ n φ φ n 1 φ n c = Θ Θ n 1 Θ n φ φ n 1 φ n / ˆΘ,

55 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 54 The transformed potential v [n+1] = v Θ 0 Θ n 0 Θ n 1 1 Θ n 0 c = v Θ 0 Θ n 0 Θ n 1 1 Θ n 0 / ˆΘ c = v + log ˆΘ x, The transformed binary eigenfunction φ [n+1] = ΩΘ, P Ωφ, P Θ φ c = ΩΘ, P Ωφ, P Θ φ / ΩΘ, P, The transformed adjoint eigenfunction ψ [n+1] = ΩΘ, P ΩΘ, ψ P ψ c = ΩΘ, P ΩΘ, ψ P ψ / ΩΘ, P, The transformed binary potential v [n+1] = v ΩΘ, P P Θ 0 c = v ΩΘ, P P Θ 0 / ΩΘ, P c = v + log ΩΘ, P x We therefore recover all of the commutative solutions given in Chapter 36 Direct verification of the solutions 361 Derivatives of quasideterminants Formulae for derivatives of quasideterminants were considered in [19] The authors consider differentiating the quasideterminant A B C d 3 Here, A is an n n matrix, d is a single entry, C is a row vector and B a column vector Differentiating both sides of 3 gives A B C d = d C A 1 B + CA 1 A A 1 B CA 1 B 33

56 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 55 The third term on right-hand side of 33 can be split into two cases Firstly, if A has the grammian-like structure, such as ΩΘ, P, then its derivative is the tensor product A = k E i F i, i=1 where E i is a column vector and F i is a row vector, both of appropriate length Therefore, the third term on the right-hand side of 33 can be written as a product of quasideterminants, giving A C B d k = d C A 1 B + CA 1 E i F i A 1 B CA 1 B 34 i=1 A B = C d + A B C 0 + k A E i A B C 0 F i 0 35 i=1 If A does not have the grammian-like structure, like the Wronskian, then it may be factorised by inserting the identity matrix expressed in the form I = k e k e T k, i=1 where e k is the column vector δ ij, which has a 1 in the kth row and a zero elsewhere If we let Ẑk be the kth row and Ẑk be the kth column of the matrix Ẑ, we have A B = d C A 1 B + C d k k CA 1 e k e T k A A 1 B CA 1 e k e T k B i=1 i=1 This gives A C B d A B = C d + k A e k A B C 0 A k B k i=1, 36 or equivalently A C B d A B = C d + k A A k C C k i=1 A B e kt 0 37 The authors of [19] then go on to show how to differentiate quasiwronskians and quasigrammians

57 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION Derivatives of quasiwronskians Let ˆΘ = θ i 1 j be the n n wronskian matrix of θ 1,, θ n, where k denotes i,j=1,,n the kth derivative, and let e k be the n-vector δ ik ie a column vector with 1 in the kth row and 0 elsewhere We will calculate derivatives of the form ˆΘ e Qi, j = n j Θ n+i 0 In this definition, i and j are allowed to take any integer values, subject to the convention that if n j lies outside the range 1,,, n, then e n j = 0 and so Qi, j = 0 There is an important special case: when n + i = n j 1 [0, n 1], ie i + j + 1 = 0 and n i < 0 we have Qi, j = Θ 0 Θ n+i 1 Θ n 1 0 Θ n+i 0 = Θ 0 Θ n+i 1 Θ n = 1 Using the same argument for n+i [0, n 1] but n+i n j 1, we see that Qi, j = 0 Assuming n is arbitrarily large, we may summarise these properties of Qi, j as 1 i + j + 1 = 0 Qi, j = 38 0 i < 0 or j < 0 and i + j If we relabel and rescale the variables so that x 1 = x, x = y, x 3 = 4t, Θ satisfies the linear equations Θ x = Θ xx, Θ x3 = Θ xxx We may allow Θ to depend on higher variables x k and impose the natural dependence Θ xk = Θx } {{ x } k Now, for any m, using the linear equations for Θ, we have ˆΘ e Qi, j = n j n x m Θ n+i+m 0 + ˆΘ e k ˆΘ Θ n+i 0 Θ k 1+m 0 k=1 e n j n 1 = Qi + m, j + Qi, kqm 1 k, j 39 k=0

58 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 57 Using the conditions 38, the above simplifies considerably and we obtain m 1 Qi, j = Qi + m, j Qi, j + m + Qi, kqm k 1, j 330 x m In particular Qi, j = Qi + 1, j Qi, j Qi, 0Q0, j, x Qi, j = Qi +, j Qi, j + + Qi, 1Q0, j + Qi, 0Q1, j, x Qi, j = Qi + 3, j Qi, j Qi, Q0, j + Qi, 1Q1, j + Qi, 0Q, j x 3 Note that these simplified formulae 330 are only valid for sufficiently large n For smaller n we should use 39 directly In addition to Qi, j we can define a shifted version, which we will call ˆQi, j: Θ 1 0 Θ ˆQi, n j 1 j = Θ n 0 Θ n+i+1 0 This satisfies equations similar to 330 k=0 363 Derivatives of quasigrammians To express the derivatives of a quasigrammian, we define Ri, j = 1 j ΩΘ, P Θ i 0 P j As we have seen in 31, solutions obtained by binary Darboux transformations are of the form v = v 0 R0, 0 As we did in the case of the quasiwronskian type of solutions we choose v 0 = 0 for simplicity Hence Θ satisfies the same linear equations as before and P, the vector of adjoint eigenfunctions, satisfies P x = P xx, P x3 = P xxx Note that choice of the trivial vacuum is inessential and direct verification can be completed for arbitrary vacuum

59 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 58 Using 34, derivatives with respect to the x m can be calculated: xm Ri, j = 1 j Ω P j Θ i+m 0 + Ω P 1m+j 1 j+m Θ i 0 m j+k Ω P k Ω Θ i 0 Θ m 1 k 0 k=0 P j m 1 = Ri + m, j Ri, j + m + Ri, krm k 1, j This final form for a derivative of a quasigrammian corresponds precisely with the formula for the quasiwronskian 330 k=0 Thus subsequent calculations carried out for the quasiwronskian solutions will be equally valid for the quasigrammian solutions To verify the quasigrammian and quasiwronskian solutions directly, we need to show that both v = Q0, 0 and v = R0, 0 are solutions of nckp In [19], the authors list some of the derivatives of v that can be substituted into nckp directly For example: v x = Q0, 0 x = [Q1, 0 Q0, 1 + Q0, 0Q0, 0], v y = Q0, 0 y = [Q, 0 Q0, + Q0, 0Q1, 0 + Q0, 1Q0, 0], v t = Q0, 0 t = [Q3, 0 Q0, 3 Q0, 0Q, 0 + Q0, 1Q1, 0 + Q0, Q0, 0] From here, using 330,we could easily calculate higher order derivatives For example v xx = [Q0, Q1, 1 + Q, 0 Q0, 0Q0, 1 + Q0, 0Q1, 0 Q0, 1Q0, 0 + Q1, 0Q0, 0 + Q0, 0Q0, 0Q0, 0] Upon substitution of v and its derivatives into 319, all the terms cancel and the solution is therefore verified 37 Matrix solutions In this section, we derive matrix solutions of nckp 319, which are new results This work, much of which is outlined in [1], builds on that given in [, 3] In [] it was shown that a matrix version of the KdV equation U t 3UU x 3U x U + U xxx = 0, 331 where U = Ux, t is a d d matrix, possessed multi-soliton solutions obtainable from Darboux transformations For example, the one-soliton solution is U = λ P sech λς υ, 33

60 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 59 where ς = 1 λ log r λ and υ = x 4λ t The solution U, as given by 33, represents a d d one-soliton matrix solution The matrix amplitude of the solution is λ P, where P = P is a projection operator sometimes referred to as the polarization of the soliton Each soliton in the matrix has phase-constant φ = 1 λ log r λ for some constant r Multisoliton solutions of a matrix sine-gordon equation were found in [36] The authors showed that this equation has quasigrammian solutions, into which they introduced projection matrices Using the quasigrammian solutions of nckp, we now follow this approach and investigate the one-, two- and three-soliton matrix solutions Taking the trivial vacuum solution v = 0 gives ΩΘ, P P v [n+1] = Θ The eigenfunctions θ i and the binary eigenfunctions ρ i satisfy θ i,xx = θ i,y, θ i,t = 4θ i,xxx and ρ i,xx = ρ i,y, ρ i,t = 4ρ i,xxx respectively The simplest nontrivial solutions of these equations are θ j = A j e η j, ρ i = B i e γ i, where η j = p j x + p j y 4p j t, γ i = q i x + q i y 4q i t and A j, B i are d m matrices At this stage the general structure of v [n+1] is ΩΘ, P v [n+1] = m m Θ d m P d m 0 d d With this choice of θ j and ρ i, = 0 d d Θ d m ΩΘ, P 1 m m P m d 334 Ωθ j, ρ i x = B T i A j e η j γ i Integrating with respect to x gives Ωθ j, ρ i = BT i A j p j q i e η j γ i + δ i,j I,

61 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 60 where δ i,j I is the constant of integration We take A j = r j P j, where r j is a scalar and P j is a projection operator With A j chosen in this way, we must have m = d and therefore 334 becomes v [n+1] = 0 d d Θ d d ΩΘ, P 1 d d P d d 335 We choose B i = I and the solution u will be a d d matrix In the case n = 1, we obtain a one-soliton matrix solution Expanding 335 gives v [] = rp e Λ I + r 1 p q eλ P, where Λ i = η i γ i When taking the inverse of Ω, we make use of the formula I ap 1 = I + ap + a P + a 3 P 3 + = I + ap + a P + a 3 P + = I + ap 1 + a + a + = I + ap 1 a 1, where a 1 is a scalar and P is any projection matrix This useful identity is subsequently used throughout We now have v [] = rp e Λ + r p q 336 Consequently, the one-soliton matrix solution is where ξ = log r p q in the x-direction and 4 detω 0 for all x,y and t, and u = v [],x = 1 p q P sech 1 Λ + ξ, 337 p 3 q 3 Each plane wave in the matrix u is travelling with speed 4 p q p 3 q 3 t in the y-direction A regular solution requires that p q r p q > 0 We have detω = 1 + If r > 0 and p > q or alternatively, if r < 0 and q > p, then t r p q eλ 338 r p q > 0 and detω is positive-definite In each case, we would obtain the same solution 337 since sech is an even function

62 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 61 In the case n = d =, we obtain a two-soliton matrix solution By expanding 335 we get v [3] = A 1 e η 1 A e η e A η j γ i j p j q i = K 1 e γ 1 K e γ Ie γ 1, Ie γ 1 + δ i,j I say Ie γ 1 Ie γ = K 1 + K, where K 1 and K satisfy K 1 I + r 1e Λ 1 P 1 p 1 q 1 K I + r e Λ P p q = e Λ 1 A 1 eλ 1 p 1 q K A 1, = e Λ A eλ p q 1 K 1 A We assume that the P j are the rank-1 projection matrices P j = µ j ν j µ j, ν j = µ jνj T µ T j ν, j where the d-vectors µ j, ν j satisfy the condition µ j, ν j 0 Solving for K 1 and K gives where K 1 = p q 1 h K = p 1 q h h p 1 q I A A 1, 339 h 1 p q 1 I A 1 A, 340 h = h 1 h p 1 q p q 1 αr 1 r, h i = e Λ i + r i, and α = µ 1, ν µ, ν 1 p i q i µ 1, ν 1 µ, ν = T rp 1P For the solution to be regular, we need detω 0 for all x, y and t Upon expanding detω, the result simplifies greatly since the trace of a projection matrix is equal to its rank and the determinant of any projection matrix is zero, and we obtain detω = 1 + κ 1 e Λ 1 + κ e Λ + κ 1 κ βe Λ 1+Λ, in which κ i = r i p i q i, i = 1, and β = 1 αp 1 q 1 p q By ordering the spectral p 1 q p q 1 parameters p > q > p 1 > q 1 and choosing r i > 0, for i = 1,, we ensure that κ i > 0 Furthermore, if we insist that α > 0, we have β > 0 and detω is therefore positivedefinite

63 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 6 We now investigate the behaviour of v [3] as t ± This will demonstrate that each soliton emerges from interaction undergoing a phase shift and that the amplitude of each soliton may also change due to the interaction For each soliton, we need to show that the asymptotic form of v [3] is the same as 336 We first fix Λ 1 by making the change of variables a similar change of variables involving y would also fix Λ 1 This gives q 3 x = ˆx p 3 1 t q 1 p 1 Λ 1 = q 1 p 1 ˆx + q 1 p 1y, Λ = q p ˆx + q p y 4 q 3 p 3 q p q1 3 p3 1 t q 1 p 1 Since Λ 1 is now independent of t, soliton 1 is at rest We may assume without loss of generality that 0 > p > q > p 1 > q 1 Then as t, and therefore v [3] r 1P 1 h 1 u 1 p 1 q 1 P 1 sech 1 Λ1 + ξ1, where ξ 1 = log r1 p 1 q 1 Note that u = v x is invariant under the transformation v v + C, where C is a constant matrix As t + we get v [3] r p 1 q p q A p q 1 A 1 + αr 1 p q p 1 q A 1 p q A h 1 r p 1 q p q 1 αr 1 r p q where + p q A r r p 1 q p q A p q 1 A 1 + αr 1 p q p 1 q A 1 p q A r p 1 q p q 1 h 1 αr 1p q p 1 q p q 1 ˆr 1 ˆP1 e Λ 1 + ˆr 1, p 1 q 1 ˆr 1 = r 1ˆµ 1, ˆν 1 µ 1, ν 1, ˆP1 = ˆµ 1 ˆν 1 ˆµ 1, ˆν 1, ˆµ 1 = µ 1 p q µ 1, ν µ p 1 q µ, ν, and ˆν 1 = ν 1 p q µ, ν 1 ν p q 1 µ, ν

64 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 63 Therefore u 1 p 1 q 1 ˆP1 sech 1 Λ1 + ξ 1 + as t, where ξ + 1 = log ˆr1 p 1 q 1 Similarly, fixing Λ gives u 1 1 p q ˆP sech Λ + ξ as t, u 1 1 p q P sech Λ + ξ + as t +, where ξ = log ˆr, ξ + p q = log r, ˆr = r ˆµ, ˆν p q µ, ν, ˆP = ˆµ ˆν ˆµ, ˆν, ˆµ = µ p 1 q 1 µ, ν 1 µ 1 p q 1 µ 1, ν 1, and ˆν = ν p 1 q 1 µ 1, ν ν 1 p 1 q µ 1, ν 1 Note that ˆµ 1,ˆν 1 µ 1,ν 1 = ˆµ,ˆν µ,ν = 1 αp 1 q 1 p q p 1 q p q 1 = β The soliton phase shifts j = ξ + j ξ j are ˆr1 1 = log r 1 = log β, = log r ˆr = log β We may now summarise the characteristics of the two-soliton matrix solution as follows: The matrix amplitude of the first soliton changes from 1 p 1 q 1 P 1 to 1 p 1 q 1 ˆP1 and the matrix amplitude of the second soliton changes from 1 p q ˆP to 1 p q P as t changes from to + If µ 1, ν = 0 P P 1 = 0 or µ, ν 1 = 0 P 1 P = 0 then α = 0 and therefore β = 1, so there is no phase shift but the matrix amplitudes may still change If µ 1, ν = 0 and µ, ν 1 = 0 giving P 1 P = P P 1 = 0 there is no phase shift or change in amplitude and so the solitons have trivial interaction

65 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 64 In general, for n 1, expanding 335 gives v [n+1] = A 1 e η 1 A e η A n e ηn e A η j γ i j p j q i = K 1 e η 1 K e η K n e ηn = Ie γ 1 Ie γ Ie γn 1 + δ i,j I n n Ie γ 1 Ie γ Ie γn 341, say, 34 n K i 343 i=1 However, for n > 3, it is very difficult to isolate each K i So we will now only investigate the three-soliton solution When n = 3, 343 gives v [4] = K 1 + K + K From we must have that K 1 = 1 A 1 K A 1 K 3A 1, 345 h 1 p 1 q p 1 q 3 K = 1 A K 1A K 3A, 346 h p q 1 p q 3 K 3 = 1 A 3 K 1A 3 K A 3, 347 h 3 p 3 q 1 p 3 q r i where h i = e Λ i + p i q i, for i = 1,, 3 Solving for K 1, K and K 3 gives h, 3 K 1 = A 1 K,3 A 1 K 3, A 1, 348 h1,, 3p q 3 p 3 q p 1 q p 1 q 3 h1, 3 K = A K 1,3 A K 3,1 A, 349 h1,, 3p 1 q 3 p 3 q 1 p q 1 p q 3 h1, K 3 = A 3 K 1, A 3 K,1 A 3, 350 h1,, 3p 1 q p q 1 p 3 q 1 p 3 q in which hi, j = p i q j p j q i h i h j r i r j α i,j, r r 3 α,3 h 1 h1,, 3 = h 1 h h 3 p q 3 p 3 q r 1 r 3 α 1,3 h p 1 q 3 p 3 q 1 r 1 r α 1, h 3 p 1 q p q 1 α 1,,3 + r 1 r r 3 p q 1 p 1 q 3 p 3 q + α 1,3,, p 1 q p q 3 p 3 q 1 K i,j = p j q i hi, j h jp i q j I A j A i,

66 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 65 for i, j {1,, 3} and i j Here, the notation from the two-soliton matrix solution has been extended to include the trace of all permutations of products of P j, j = 1,, 3, so that α i,j = T rp i P j = µ j, ν i µ i, ν j µ i, ν i µ j, ν j = T rp jp i, α 1,,3 = T rp 1 P P 3 = µ 1, ν 3 µ 3, ν µ, ν 1 µ 1, ν 1 µ, ν µ 3, ν 3 = T rp P 3 P 1 = T rp 3 P 1 P, α 1,3, = T rp 1 P 3 P = µ 3, ν 1 µ, ν 3 µ 1, ν µ 1, ν 1 µ, ν µ 3, ν 3 = T rp P 1 P 3 = T rp 3 P P 1 Substituting into 344 gives v [4] = h1,, 3 b,3a 1 + b 1,3 A + b 1, A 3 + b 1,3, A 1 A + b 1,,3 A 1 A 3 +b,3,1 A A 1 + b,1,3 A A 3 + b 3,,1 A 3 A 1 + b 3,1, A 3 A, 351 in which hi, j b i,j := and p i q j p j q i r j α i,j,k b i,j,k := h j, p k q j p j q i α i,k p k q i if α i,k 0 For the solution to be regular, we need detω 0 for all x, y and t Here we have Ω = A j e η j γ i p j q i + δ i,j I 3 3 Using the fact that T ra j = r j and deta j = 0, for j = 1,, 3, expanding detω gives detω = 1 + κ 1 e Λ 1 + κ e Λ + κ 3 e Λ 3 + κ 1 κ β 1, e Λ 1+Λ + κ κ 3 β,3 e Λ +Λ 3 + κ 1 κ 3 β 1,3 e Λ 1+Λ 3 + κ 1 κ κ 3 β 1,,3 e Λ 1+Λ +Λ 3, 35 where β 1,,3 = 1 α 1,p 1 q 1 p q α,3p q p 3 q 3 p 1 q p q 1 p q 3 p 3 q +p 1 q 1 p q p 3 q 3 β i,j = 1 α i,jp i q i p j q j p i q j p j q i κ i = r i for i = 1,, 3 p i q i α 1,3p 1 q 1 p 3 q 3 p 1 q 3 p 3 q 1 α 1,,3 p q 1 p 3 q p 1 q 3 + α 1,3, p 1 q p q 3 p 3 q 1 for i, j {1,, 3}, i j,,

67 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION a t << 0 b t >> 0 Figure 31: The chosen configuration of the three-soliton solution Ordering the spectral parameters p 3 > q 3 > p > q > p 1 > q 1 guarantees that β i,j > 0 and κ i > 0 From 35, detω will be positive-definite if α 1,,3 p q 1 p 1 q 3 p 3 q + α 1,3, p 1 q p q 3 p 3 q 1 > 0 To determine the asymptotic forms of each matrix soliton, we fix each Λ i, i = 1,, 3 and assume without loss of generality that 0 > p 3 > q 3 > p > q > p 1 > q 1 In doing so, the solution has the configuration detailed in Figure 31 For soliton 1, as t, h i + for i =, 3 Then we can see from that K i 0 for i =, 3 This may be compared with similar expressions in the two-soliton matrix solution, giving v [4] r 1 P 1 h 1, where r 1 = r 1, P 1 = µ 1 ν 1 µ 1,ν 1, µ 1 = µ 1 and ν 1 = ν 1 So as t, we have u 1 p 1 q 1 P 1 sech 1 Λ 1 + ξ 1, where ξ 1 = log r 1 p 1 q 1 As t +, h i r i p i q i, for i =, 3 Using the fact that u is invariant under the transformation v [4] v [4] + C, where C is a constant matrix, we have that v [4] r 1 + P 1 +, e Λ r p 1 q 1

68 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 67 where = r 1β 1,,3, P 1 + β = µ+ 1 ν+ 1,3 µ + 1, ν+ 1, µ + 1 = µ 1 + p q p 3 q 3 µ 1, ν 3 µ 3, ν µ, ν β,3 µ 3, ν 3 p 1 q 3 p 3 q µ 1, ν p 1 q p 3 q 3 + p q 3 p 3 q µ 1, ν µ, ν 3 µ 3, ν 3 β,3 µ, ν p 1 q p q 3 µ 1, ν 3 µ 3, p 1 q 3 p q ν 1 + = ν 1 + p q p 3 q 3 µ 3, ν 1 µ, ν 3 µ, ν β,3 µ 3, ν 3 p q 3 p 3 q 1 µ, ν 1 p q 1 p 3 q 3 + p q 3 p 3 q µ, ν 1 µ 3, ν µ 3, ν 3 β,3 µ, ν p 3 q p q 1 µ 3, ν 1 ν 3 p 3 q 1 p q r + 1 = r 1µ + 1, ν+ 1 µ 1, ν 1 µ ν So we have u 1 p 1 q 1 P + 1 sech 1 Λ 1 + ξ + 1 as t +, where ξ 1 + = log r + 1 p 1 q 1 Fixing Λ brings soliton to rest As t, h 1 r 1 p 1 q 1 and h 3 + This gives where v [4] K 1, + K,1 r P, e Λ r + p q r = r µ, ν µ, ν = r β 1,, P = µ ν µ, ν, µ = µ p 1 q 1 µ, ν 1 p q 1 µ 1, ν 1 µ 1, and ν = ν p 1 q 1 µ 1, ν p 1 q µ 1, ν 1 ν 1 As t +, h 1 + and h 3 r 3 p 3 q 3 This gives where v [4] K,3 + K 3, r + P +, e Λ r + + p q r + = r µ +, ν+ µ, ν = r β,3, P + = µ+ ν+ µ +, ν+, µ + = µ p 3 q 3 µ, ν 3 p q 3 µ 3, ν 3 µ 3, and ν + = ν p 3 q 3 µ 3, ν p 3 q µ 3, ν 3 ν 3 So the asymptotic forms for soliton are u 1 p q P sech 1 Λ + ξ u 1 p q P + sech 1 Λ + ξ + as as t, t +,

69 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 68 where ξ = log r p q and ξ + = log r + p q With Λ 3 fixed, soliton 3 is a rest As t, h i r i p i q i for i = 1, This gives where v [4] r3 P 3, e Λ r p 3 q 3 r 3 = r 3µ 3, ν 3 µ 3, ν 3 = r 3β 1,,3, P3 β = µ 3 ν 3 1, µ 3, ν 3, µ 3 = µ 3 + p q p 1 q 1 µ 3, ν 1 µ 1, ν µ, ν β 1, µ 1, ν 1 p 3 q 1 p 1 q µ 3, ν p 3 q p 1 q 1 + p q p 1 q 1 µ 3, ν µ, ν 1 µ 1, ν 1 β 1, µ, ν p 3 q p q 1 µ 3, ν 1 µ 1, p 3 q 1 p q ν3 = ν 3 + p q p 1 q 1 µ 1, ν 3 µ, ν 1 µ, ν β 1, µ 1, ν 1 p q 1 p 1 q 3 µ, ν 3 p q 3 p 1 q 1 + p q p 1 q 1 µ, ν 3 µ 1, ν µ 1, ν 1 β 1, µ, ν p 1 q p q 3 µ 1, ν 3 ν 1 p 1 q 3 p q As t +, h i +, for i = 1, This gives µ ν v [4] r 3 + P 3 +, e Λ r p 3 q 3 where r + 3 = r 3, P + 3 = µ+ 3 ν+ 3 µ + 3,ν+ 3, µ+ 3 = µ 3 and ν + 3 = ν 3 So the asymptotic forms for soliton 3 are u 1 p 3 q 3 P 3 sech 1 Λ 3 + ξ 3 u 1 p 3 q 3 P + 3 sech 1 Λ 3 + ξ + 3 as as t, t +, where ξ 3 = log r 3 p 3 q 3 and ξ 3 + = log r + 3 p 3 q 3 The soliton phase shifts j = ξ j + ξ j are µ + 1 = log 1, ν 1 + µ 1, ν 1 = log µ + = log, ν + µ, ν = log µ + 3 = log 3, ν 3 + µ 3, ν 3 38 Plots of the matrix solutions β1,,3 β,3 β,3 β 1, β1, = log β 1,,3,, In this section, we demonstrate the interaction properties of the two-soliton matrix solution of nckp with various plots

70 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 69 where Figure 3 shows a plot of the generic two-soliton matrix solution u = u ij, i, j = 1,, r 1 = ˆr 1 = 40763, ˆr = 0381 r = 1, µ 1 = 1 0 ˆµ 1 = , µ = ˆµ = , ˆν 1 = ν 1 = , ˆν = ν = 0333, P 1 = 1 ˆP = 0901, ˆP = P = 013, as t changes from to + This plot shows both a change in matrix amplitude and a phase-shift upon interaction Figure 33 shows a plot of the two-soliton matrix solution u = u ij, i, j = 1,, where r 1 = ˆr 1 = r 1, ˆr = 1 r = ˆr, µ 1 = 1 3 ˆµ 1 = µ 1, µ = ˆµ = , ˆν 1 = ν 1 = , ˆν = ν = ˆν, 0 P 1 = 04 ˆP = 0344, ˆP = P = 019, as t changes from to + phase-shift because µ 1, ν = 0 This plot shows a change in matrix amplitude but no Figure 34 shows a plot of the two-soliton matrix solution u = u ij, i, j = 1,, where

71 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 70 r 1 = ˆr 1 = r 1, ˆr = 1 r = ˆr, µ 1 = 1 3 ˆµ 1 = µ 1, µ = 4 15 ˆµ = µ, ˆν 1 = ν 1 = ˆν 1, ˆν = ν = ˆν, 0143 P 1 = 0381 ˆP 1 = P 1, ˆP = 0381 P = P, as t changes from to + This plot shows no change in matrix amplitude and no phase-shift because µ 1, ν = 0 = µ 1, ν

72 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 71 a b Figure 3: a Plot of matrix KP two-soliton interaction at t = 0 with parameters given by p 1 = 1 4, p = 19, q 1 = 39 two-soliton interaction with P j = r j = 1 for j = 1, and q = 1 b Plot of the corresponding scalar KP

73 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 7 Figure 33: Plot of matrix KP two-soliton interaction at t = 0 with parameters given by p 1 = 1 4, p = 19, q 1 = 39 and q = 1

74 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 73 Figure 34: Plot of matrix KP two-soliton interaction at t = 0 with parameters given by p 1 = 1 4, p = 19, q 1 = 39 and q = 1 39 Reduction to matrix KdV To make the reduction from matrix KP solutions to matrix KdV solutions, we set p i = q i = λ i For the one-soliton solution, this gives Λ = λx 4λ t So v [] = rp e λx 4λt + r, λ

75 CHAPTER 3 A NONCOMMUTATIVE KP EQUATION 74 giving u = λ P sech λυ + ς, where υ = x 4λ t and ς = 1 λ log r λ For the two-soliton matrix solution, we have h i = e λ iυ i + r i λ i for i = 1, The asymptotic forms for soliton 1 are u λ 1P 1 sech λ 1 υ 1 + ς 1 as t, u λ 1 ˆP 1 sech λ 1 υ 1 + ς + 1 as t +, in which υ 1 = x 4λ 1t, ς1 = log r1, ς 1 + λ = log ˆr1, 1 λ 1 ˆr 1 = r 1ˆµ 1, ˆν 1 µ 1, ν 1, P 1 = µ 1 ν 1 µ 1, ν 1, ˆP1 = ˆµ 1 ˆν 1 ˆµ 1, ˆν 1, ˆµ 1 = µ 1 λ µ 1, ν µ λ 1 + λ µ, ν, and ˆν 1 = ν 1 λ µ, ν 1 ν λ 1 + λ µ, ν The asymptotic forms for soliton are u λ ˆP sech λ υ + ς as t, u λ P sech λ υ + ς + as t +, in which υ = x 4λ t, ς = log ˆr, ς + λ = log r, λ ˆr = r ˆµ 1, ˆν 1 µ 1, ν 1, P 1 = µ ν µ, ν, ˆP = ˆµ ˆν ˆµ, ˆν, ˆµ = µ λ µ, ν 1 µ 1 λ 1 + λ µ 1, ν 1, and ˆν = ν λ 1µ 1, ν ν 1 λ 1 + λ µ 1, ν 1 These results from the reduction match up with those given in []

76 Chapter 4 A noncommutative mkp equation Noncommutative mkp equations have been considered by Hamanaka and Toda [7], and Wang and Wadati [51] in the case where the noncommutativity arises through the independent variables This suggests that we may be able to follow the methods of the previous chapter and try to: derive an ncmkp equation, two families of solutions of ncmkp that can be expressed as quasiwronskians and quasigrammians, directly verify the solutions and investigate matrix solutions We will also look at an nc Miura transformation which maps solutions of ncmkp to solutions of nckp 41 A noncommutative mkp hierarchy A noncommutative mkp hierarchy has been developed by Kupershmidt in [34], but in a more algebraic setting In [51], a ncmkp hierarchy is given that uses analytic Hirota bilinear identities to give the hierarchy in a condensed form Here, we construct the ncmkp hierarchy in the spirit of Sato theory With L = L mkp as defined by 36, we obtain the differential operators P 1 L = x, P 1 L = x + w x, 41 P 1 L 3 = x 3 + 3w x + 3w x + w + w 1 x, P 1 L 4 = x 4 + 4w x 3 + 6w x + 4w 1 + 6w x + 4w 3 + 6w 1x + 4w xx + 4w + 8ww x + 4w x w + 6ww 1 + 6w 1 w x, 75

77 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 76 which, via the evolution equation 37, give the ncmkp hierarchy: w x1 = w x, w 1x1 = w 1x, L x1 = [P 1 L, L] w x1 = w x,, 4 L x = [P 1 L, L] w y = w xx + w 1x + ww x + [w, w 1 ], w 1y = w 1xx + w x + w 1 w x + ww 1x + [w, w ], w y = w xx + w 3x + ww x + 4w w x w 1 w xx + [w, w 3 ], w 3y = w 3xx + w 4x + ww 3x + 6w 3 w x w 1 w xxx 6w w xx + [w, w 4 ],, 43 L x3 = [P 1 L 3, L] L x4 = [P 1 L 4, L] w t = w xxx + 3w 1xx + 3w x + 6ww 1x + 3w 1 w x + 3w x w 1 + 3ww xx + 3wx + 3w w x + 3[w, w 1 ] + 3[w, w ],, w x4 = w xxxx + 6w xx + 4w 1xxx + 4w 3x + 4ww xxx +6w x w xx + 4w xx w x + 4w x ww x + 6w w xx +8wwx + 4w 3 w x + 1ww 1xx + 1w x w 1x +6w 1x w x + 6w 1x w 1 + 6w 1 w 1x + w 1 w xx +4w xx w 1 + 1w w 1x + 8ww x w 1 + 6ww 1 w x +4w x ww 1 + 4w 1 ww x + w 1 w x w + 1ww x +6w w x + 6w x w + 6[w, w1 ] + 6[w, w ] [w 3, w 1 ] + 4[w, w 3 ], The term [w, w 1 ] in the first component of 43 prevents us from recursively expressing the fields w 1, w, in terms of w and its x- and y-derivatives However, using the second component of 43 and the first component of 44, we obtain 0 = w t w xxx 3w 1xx 6ww 1x 3w 1y 6w x w 1 6ww xx 6w x 6w w x 6[w, w 1 ] 46

78 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 77 To eliminate the field w 1, we make the change of variables w 1 = 1 w x + w W Thus, from the first component of 43, and from 46, we obtain the following equations: w t + w xxx 6ww x w + 3W y + 3[w x, W ] + 3[w xx, w] 3[W, w ] = 0, 47 W x w y + [w, W ] = 0, 48 where the scaling t 4t has been made Equations 47 and 48 could also be obtained through the Lax pair L mkp = x + w x y, M mkp = 4 x 3 + 1w x + 6w x + w + W x + t The compatibility condition [L mkp, M mkp ] = 0 gives 47 and 48, which represent the ncmkp equation in a slightly different but equivalent form to that of Wang and Wadati in [51] Unlike the commutative mkp equation, equation 48 is not satisfied by introducing a potential Instead, we follow the approach in [51] by letting w = f x f 1, and W = f y f 1, where f = fx, x q is a differentiable function with an inverse It is assumed that the function f and its x- and x q -derivatives do not, in general, commute These choices of w and W satisfy equation 48 It is important to reiterate here that w log f x and W log f y Now that we have w = f x f 1 and W = f y f 1, we can attempt to eliminate w 1, w, from the hierarchy commute, we have seen that When w, w 1, w, w 3, and their x- and x q -derivatives w y = w x + w + w 1 x, w t = w xx + 3w 1x + 3w + 3ww x + w 3 + 6ww 1 x, w x4 = w xxx + 6w x + 4w 1xx + 4w 3 + w 4 + 1ww + 6w w x + 1w w 1 + 6w1 + 6w x w 1 + 1ww 1x + 3wx + 4ww xx x Since w = V x, for the first three fields, say, we can write V xy = w x + w + w 1 x, V xt = w xx + 3w 1x + 3w + 3ww x + w 3 + 6ww 1 x, V xx4 = w xxx + 6w x + 4w 1xx + 4w 3 + w 4 + 1ww + 6w w x + 1w w 1 + 6w1 + 6w x w 1 + 1ww 1x + 3wx + 4ww xx x

79 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 78 Integrating both sides of each of these equations with respect to x and isolating w 1, w, w 3 gives w 1 = 1 V y 1 w x 1 w, w = 1 3 V t 1 3 w xx w 1x ww x 1 3 w3 ww 1, w 3 = 1 4 V x w xxx 3 w x w 1xx 1 4 w4 3ww 3 w w x 3w w 1 3 w 1 3 w xw 1 3ww 1x 3 4 w x ww xx Furthermore, we have that f x f 1 log f x V x and f y f 1 log f y V y However, in the noncommutative case we have w = f x f 1 We may therefore take w 1, w, w 3, to be of the form w 1 = a 1 w x + a w + a 3 f y f 1, w = b 1 w xx + b w 1x + b 3 ww x + b 4 w x w + b 5 w 3 + b 6 ww 1 + b 7 w 1 w + b 8 f t f 1, w 3 = c 1 w xxx + c w x + c 3 w 1xx + c 4 w 4 + c 5 ww + c 6 w w + c 7 w w x + c 8 w x w + c 9 ww x w + c 10 w w 1 + c 11 w 1 w + c 1 ww 1 w + c 13 w1 + c 14 w x w 1 + c 15 w 1 w x + c 16 ww 1x + c 17 w 1x w + c 18 wx + c 19 ww xx + c 0 w xx w + c 1 f x4 f 1,, where a n, b n, c n,, n = 1,, 3,, are constants to be chosen such that the resulting noncommutative fields w 1, w, w 3 will then satisfy the ncmkp hierarchy We know that, for n = 1,, 3, w xn = f x f 1 xn = f xxn f 1 + f x f 1 f xn f 1 = f xxn f 1 wf xn f 1 49 Using the terms w y, w 1y, w y, in 43, we can calculate w 1, w, w 3, For example, w y = 1 + a 1 w xx + + a 1 + a ww x + a a 1 w x w + a 3 wf y f 1 + a 3 f xy f 1 For 49 to be satisfied by w 1 and with w = f x f 1, we require that a 1 = 1, a = 1 and a 3 = 1

80 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 79 Using the same approach for w and w 3, we require that b 1 = 1 3, b = 1, b 3 = 3, b 4 = 1 3, b 5 = 1 3, b 6 = 1, b 7 = 1, b 8 = 1 3, c 1 = 1 4, c = 3, c 3 = 1, c 4 = 1 4, c 5 = 3, c 6 = 3, c 7 = 3 4, c 8 = 1 4, c 9 = 1, c 10 = 1, c 11 = 1, c 1 = 1, c 13 = 3, c 14 = 1, c 15 = 1, c 16 =, c 17 = 1, c 18 = 3 4, c 19 = 3 4, c 0 = 1 4, c 1 = 1 4 Therefore, the fields w 1, w, w 3 are w 1 = 1 w x 1 w 1 f yf 1, w = 1 3 w xx w 1x 3 ww x 1 3 w xw 1 3 w3 ww 1 w 1 w 1 3 f tf 1, w 3 = 1 4 w xxx 3 w x w 1xx 1 4 w4 3 ww 3 w w 3 4 w w x 1 4 w xw 1 ww xw w w 1 w 1 w ww 1 w 3 w 1 w x w 1 1 w 1w x ww 1x w 1x w 3 4 w x 3 4 ww xx 1 4 w xxw 1 4 f x 4 f 1 Although tedious, this procedure could easily be used to obtain w 4, w 5, these fields in terms of f gives Rewriting w 1 = 1 f xxf 1 f x f 1 f x f 1 1 f yf 1, 410 w = 1 6 f xxxf 1 f x f 1 f x f 1 f x f f xf 1 f xx f 1 + f xx f 1 f x f f xyf 1 f y f 1 f x f 1 1 f xf 1 f y f f tf 1, 411 w 3 = 1 4 f x 4 f f xtf f xxyf 1 f t f 1 f x f 1 1 f xf 1 f t f f xf 1 f xy f f yf 1 f y f f yf 1 f xx f f xxf 1 f y f f xxf 1 f xx f f xyf 1 f x f 1 1 f xxxf 1 f x f f xf 1 f x f 1 f y f f xf 1 f x f 1 f xx f 1 3 f xf 1 f y f 1 f x f f xf 1 f xx f 1 f x f 1 3f y f 1 f x f 1 f x f 1 + 3f xx f 1 f x f 1 f x f 1 6f x f 1 f x f 1 f x f 1 f x f 1 41 This hierarchy, in terms of f, was derived from a different perspective by Dimakis and Müller-Hoissen in [8] in what they refer to as a functional representation of the hierarchy

81 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 80 4 Quasiwronskian solutions obtained from Darboux transformations In this section, we look at quasiwronskian solutions of ncmkp obtained from the Darboux tranformation G θ = θ 1 x 1 x θ 1 = 1 θθ x 1 x We will use the pseudodifferential operator L mkp Let θ i, i = 1,, n be a particular set of eigenfunctions and introduce the notation Θ = θ 1,, θ n It is again assumed that the eigenfunction and its derivatives do not commute Let φ = φ [1] be an eigenfunction of L mkp[1] = L mkp and θ [1] = θ 1 Then φ [] := G θ[1] [φ [1] ] and θ [] = φ [] φ θ are eigenfunctions for L mkp[] = G θ[1] L mkp G 1 θ [1] In general, for n 1 we define the nth Darboux transform of φ by φ [n+1] = φ [n] θ [n] θ [n]x 1 φ [n]x, 413 in which θ [k] = φ [k] φ θk It can be shown by induction that φ [n+1] as given by 413 can be expressed as φ [n+1] = Θ φ Θ n 1 φ n 1 Θ n φ n 414 In the initial case n = 1, 413 gives φ [] = φ θ 1 θ 1 1,x φ x = θ 1 θ 1,x φ φ x So the result is true for n = 1 Substituting n + 1 for n in 413 gives φ [n+] = φ [n+1] θ [n+1] θ [n+1]x 1 φ [n+1]x 415

82 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 81 Using 36 and 310, we have Θ 1 θ 1 n+1 θ 1 [n+1] = Θ 1 θ 1 n+1 n + k=1 Θ n θ n n+1 Θ 0 Θ 1 Θ n+1 0 = Θ n 1 0 Θ Θ n 1 n Θ 0 e k Θ 1 Θ n θ n+1 n+1 Θ 1 θ 1 n+1 θ n n+1 Θ k+1 θ k+1 n+1 Θ 1 θ 1 n+1 Θ n θ n n Similarly Θ 0 Θ 1 Θ n+1 0 φ 1 [n+1] = Θ 1 φ 1 Θ n 1 0 Θ n φ n Θ n 1 φ n Substituting 416 and 417 into 415 and using the nc Jacobi identity 35 gives φ [n+] = = Θ φ Θ θ n+1 Θ 1 θ 1 1 n+1 Θ 1 φ 1 φ n 1 Θ n 1 θ n 1 n+1 Θ n θ n n+1 φ n Θ n θ n Θ n φ n 418 n+1 Θ n+1 θ n+1 n+1 Θ n+1 φ n+1 Θ θ n+1 φ θ n n+1 φ n 419 Θ n 1 Θ n Θ n Θ n+1 θ n+1 n+1 φ n+1 This proves the inductive step, completing the proof of 414

83 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 8 We can determine the Darboux-transformed fields w, w 1, w, by calculating L mkp = θ 1 1 x x θ 1 Lθ 1 θ 1 x = x θθx 1 f x θθx 1 f 1 + θθ 1 x +, x 1 θθ 1 x f x θθ 1 f 1 θθ 1 f x θθ 1 x x x f xx θθ 1 x f 1 f 1 1 θθ 1 x f 1 y θθx 1 f 1 x 1 which preserves the structure of the ncmkp hierarchy The coefficients w = θθx 1 f x θθx 1 f 1, w 1 = 1 θθ 1 x f xx θθ 1 f 1 θθ 1 1 θθ 1 x f 1 y θθx 1 f 1, x x f x θθ 1 x f 1 θθ 1 f x θθ 1 x x f 1 will satisfy 43 and 44 In particular, w, which could also be obtained from L mkp = G θ L mkp G 1 θ or M mkp = G θ M mkp G 1 θ, will satisfy the ncmkp equation Using the fact that w is of the form f x f 1, we obtain f = θθx 1 θ 0 f = θ x 1 f 40 It can be proved by induction that after n Darboux transformations, we have, for n 1 Θ 0 f [n+1] = Θ n 1 0 Θ n 1 f 41 When n = 1, the result 41 follows directly from 40 Upon substituting n + 1

84 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 83 for n, using 416, the nc Jacobi identity 35 and the homology relations 36 we have f [n+] = θ [n+1] θ [n+1],x 1 f Θ n+1 θ n+1 1 n+1 Θ 0 Θ θ n+1 Θ 1 θ n+1 1 Θ 1 0 = Θ n 1 θ n 1 f n+1 Θ n θ n Θ n 1 θ n 1 n+1 Θ n 1 0 n+1 Θ n θ n n+1 Θ n 1 Θ n+1 θ n+1 1 n+1 Θ θ n+1 Θ 1 θ 1 n+1 = Θ n 1 θ n 1 f n+1 Θ n θ n Θ n 1 θ n 1 n+1 n+1 Θ n θ n n+1 Now if we use the quasi-plücker coordinate formula 31, we get f [n+] = This completes the proof of 41 Θ θ n+1 0 θ n f n+1 0 θ n+1 n+1 1 Θ n Θ n+1 An analogous transformation can be made on f 1 Let g = f 1 This gives w = g 1 x g = g 1 g x The effect of is that L mkp = G θ L mkp G 1 θ, LmKP = G θ L mkp G 1 θ or MmKP = G θ M mkp G 1 θ w = g 1 g x = gθ x θ 1 1 gθ x θ 1 x, giving g = gθ x θ 1 = g θ 1 θ x 0

85 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 84 After n Darboux transformations we have Θ 1 Θ 1 0 g [n+1] = g 4 Θ n 1 0 Θ n 0 We can see that 4 is consistent with 41 since Θ 1 Θ 1 0 Θ n 1 0 Θ n 0 1 Θ 0 = Θ n 1 0 Θ n 1 43 Quasigrammian solutions obtained from binary Darboux transformations In the same way as for nckp, we may extend the notion of the adjoint to obtain the adjoint Lax pair L mkp = x w x w x + y, M mkp = 4 3 x + 1w x + 63w x w W x + 6w xx w xw w w x W x t The compatibility condition [L mkp, M mkp] gives w t + w xxx 6w w xw + 3W y + 3[w x, W ] + 3[w xx, w ] 3[W, w ] = 0, 43 W x w y + [w, W ] = 0, 44 which is the adjoint of 47 and 48 In order to be able to define a binary Darboux transformatiom, we need to introduce a potential Ωφ, ψ satisfying Ωφ, ψ x = ψ φ x, Ωφ, ψ y = ψ wφ x + ψ φ xx ψ xφ x, Ωφ, ψ t = ψ xxφ x ψ φ xxx + ψ xφ xx 3ψ w φ x 3ψ W φ x 3ψ w x φ x + 6ψ xwφ x 6ψ wφ xx

86 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 85 The parts of this definition are compatible when L mkp [φ] = M mkp [φ] = 0 and L mkp[ψ] = M mkp[ψ] = 0 In addition, we may define ΩΦ, Ψ for any row vectors Φ and Ψ such that L mkp [Φ] = M mkp [Φ] = 0 and L mkp[ψ] = M mkp[ψ] = 0 If Φ is an n-vector and Ψ is an m-vector then ΩΦ, Ψ is an m n matrix Let θ i, i = 1,,, n be the eigenfunctions defined in the previous section and let ρ i, i = 1,,, n be adjoint eigenfunctions For Lax operators with matrix coefficients, a binary Darboux transformation was defined in [41] and is φ [n+1] = φ [n] θ [n] Ωρ [n], θ [n] 1 Ωρ [n], φ [n] and ψ [n+1] = ψ [n] ρ [n] Ωρ [n], θ [n] 1 Ωψ [n], θ [n] in which θ [n] = φ [n] φ θn, ρ [n] = ψ [n] ψ ρn Using the notation Θ = θ 1, θ n and P = ρ 1,, ρ n we have, for n 1 ΩΘ, P Ωφ, P φ [n+1] = Θ φ, 45 ΩΘ, P ψ [n+1] = ΩΘ, ψ P ψ 46 If the above binary Darboux transformation holds then, using 35, we have ΩΘ, P Ωφ, P Ωφ [n+1], ψ [n+1] = ΩΘ, ψ Ωφ, ψ To prove 45, first observe that when n = 1, φ [] = φ θ 1 Ωρ 1, θ 1 1 Ωρ 1, φ Ωρ = 1, θ 1 Ωφ, ρ 1 φ θ 1

87 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 86 Using the nc Jacobi identity 35 and the row homological relations 36, we have φ [n+] = φ [n+1] θ [n+1] Ωρ [n+1], θ [n+1] 1 Ωρ [n+1], φ [n+1] ΩΘ, P Ωφ, P = Θ φ 1 ΩΘ, P Ωθ n+1, P Ωθ, P Ωφ, P ΩΘ, P Ωφ, P Θ θ n+1 ΩΘ, ρ Ωθ, ρ ΩΘ, ρ Ωφ, ρ ΩΘ, P Ωφ, P = Θ φ 1 ΩΘ, P Ωθ n+1, P Ωθ, P Ωφ, P ΩΘ, P Ωφ, P Θ θ n+1 ΩΘ, ρ Ωθ, ρ ΩΘ, ρ Ωφ, ρ ΩΘ, P Ωθ n+1, P Ωφ, P = ΩΘ, ρ n+1 Ωθ n+1, ρ n+1 Ωφ, ρ n+1 Θ θ n+1 φ This proves the inductive step and completes the proof The proof of 46 is very similar The effect of is that ˆL mkp = G θ,φx L mkp G 1 θ,φ x, ˆLmKP = G θ,φx L mkp G 1 θ,φ x or ˆMmKP = G θ,φx M mkp G 1 θ,φ x ˆf = 1 θω 1 ρ Ω ρ f = f 47 θ 1 After n Darboux transformations we have, for n 1 ΩP, Θ P f [n+1] = f 48 Θ I Proof of 48 is again by induction For n = 1, the result clearly follows from 47 Next, replacing n with n + 1 gives f [n+] = I Θ [n+1] ΩΘ [n+1], P [n+1] 1 P [n+1] f[n+1] ΩΘ, P Ωθ = I n+1, P ΩΘ, P Ωθ n+1, P Θ θ n+1 ΩΘ, ρ n+1 Ωθ n+1, ρ n+1 ΩΘ, P P ΩP, Θ P ΩΘ, ρ n+1 ρ n+1 Θ I f 1

88 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 87 By observing that ΩΘ, P P ΩP, Θ P ΩΘ, ρ n+1 ρ n+1 Θ I = ρ n+1 ΩΘ, ρ n+1ωθ, P 1 P I ΘΩP, Θ 1 P ΩΘ, P P = ΩΘ, ρ n+1 ρ n+1 and using the nc Jacobi identity 35 we have ΩP, Θ P f [n+] = Θ I ΩΘ, P Ωθ n+1, P ΩΘ, P Ωθ n+1, P Θ θ n+1 ΩΘ, ρ n+1 Ωθ n+1, ρ n+1 ΩΘ, P P ΩΘ, ρ n+1 ρ f n+1 ΩP, Θ ΩP, θ n+1 P = Ωρ n+1, Θ Ωρ n+1, θ n+1 ρ n+1 f, Θ θ n+1 I which proves the inductive step and the proof is now complete There appears to be no way of inverting 48 Consequently, an analogous transformation on f 1 [n+1] is not made in the quasigrammian case 1 44 Reduction to commutative Wronskian and Grammian solutions As we saw with nckp, all of the quasideterminants expressing the Darboux-transformed eigenfunctions and variables f [n+1] should reduce to the corresponding commutative results in Chapter Using 311, in the commutative case we have: The transformed eigenfunction φ [n+1] = Θ Θ n 1 Θ n φ φ n 1 φ n c = 1 n+1 Θ Θ n 1 Θ n φ φ n 1 φ n / ˆΘ,

89 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 88 The transformed variable f [n+1] = Θ 0 Θ n 1 0 Θ n 1 f c = 1 n+1 e V Θ 0 Θ n 1 0 Θ n 1 / Θ 1 Θ n This gives V [n+1] c = logf[n+1] c = V + 1 n log Θ 0 Θ n 1 0 Θ n 1 / Θ 1 Θ n c = V + 1 n log ˆΘ / Θ 1 Θ n The transformed binary eigenfunction φ [n+1] = ΩΘ, P Ωφ, P Θ φ c = ΩΘ, P Ωφ, P Θ φ / ΩΘ, P, The transformed adjoint eigenfunction ψ [n+1] = ΩΘ, P ΩΘ, ψ P ψ c = ΩΘ, P ΩΘ, ψ P ψ / ΩΘ, P, The transformed binary variable f [n+1] = ΩΘ, P P Θ I f c = e V ΩΘ, P P Θ I / ΩΘ, P Therefore V [n+1] c = logf[n+1] c = V + log ΩΘ, P x We therefore recover all of the commutative solutions given in Chapter

90 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION Direct verification of the solutions Since we did not derive an expression for only prove the quasiwronskian solutions in this section 1 ˆf [n+1] when finding quasigrammian solutions, we The Lax pairs of KP and mkp are the same when the vacuum solutions are trivial Let Θ be a common eigenfunction for these two trivial vacuum Lax pairs For ncmkp, the trivial vacuum solution, obtained from f = 1, which gives w = 0 = W, is Θ 0 Θ 1 0 F = 49 Θ n 1 0 Θ n 1 To verify directly that 49 is a solution, we also use the solutions v = Q and ˆv = ˆQ of nckp 319 Here, for convenience this is written in potential form Θ 0 Θ 1 0 Θ 1 0 Θ 0 Q = Q0, 0 =, ˆQ = ˆQ0, 0 = Θ n 1 1 Θ n 1 Θ n 0 Θ n+1 0 Note that ˆQ is only a solution if the vacuum is zero In a similar way we define Θ 0 Θ 1 0 F j =, 430 Θ n j 1 Θ n 0 so that F = F 0 Using 314, we have homological relations expressed as the identities F Q0, j = F j From the quasi-plücker coordinates 31, the inverse of F can be obtained from the expression for F by swapping the boxed entry and the 1 in the last column of F Thus we

91 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 90 define and additionally Then Θ 1 Θ 1 0 G = F 1 = Θ n 1 0 Θ n 0 Θ 1 Θ 1 0 Gj = Θ n 1 0 Θ n+j 0 43 ˆQj, 0G = Gj Now consider the derivatives of F j: using 430 and 431, F j x = F ˆQ0, j F j + 1 = F ˆQ0, j Q0, j 434 More generally, if we assume that Θ satisfies the linear equations Θ xk = Θx } {{ x, we have } k F j xk+1 = Thus, using 330, = F k F i ˆQk i, j F k + j i=0 ˆQk, j + k Q0, i 1 ˆQk i, j Q0, k + j 436 i=1 F x = F ˆQ F 1 = F ˆQ Q, 437 F xx = F ˆQ Q + ˆQ x Q x, 438 and and so F y = F ˆQ1, 0 + F 1 ˆQ F, F xx + F y = F ˆQ x 439

92 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 91 Using the nc Jacobi identity and 433 we can show that ˆQ0, 1 = Θ 1 0 Θ n 1 1 Θ n 0 Θ n+1 0 = Θ 1 0 Θ Θ n Θ n 0 0 Θ n = Q1, 0 G1F 1 Q = Q1, 0 + ˆQQ This is the noncommutative version of the first bilinear identity in the ncmkp hierarchy It can be generalized to get to the other members of the hierarchy: ˆQi, j = Qi + 1, j 1 + ˆQi, 0Q0, j This follows immediately from considering ˆQi, j written as Θ 1 0 Θ Θ n j 0 1 Θ n Θ n 0 0 Θ n+1+i 0 0 = Θ 0 Θ 1 0 Θ n j 1 Θ n 1 0 Θ n+1+i 0 Θ 1 Θ 1 0 Θ n j 0 Θ n 1 0 Θ n+1+i 0 Θ 1 Θ 1 0 Θ n j 0 Θ n 1 0 Θ n 0 1 Θ 0 Θ 1 0 Θ n j 1 Θ n 1 0 Θ n 0 = Qi + 1, j 1 Gi + 1F 1 Q0, j 1 and then using 433 On substituting F and its derivatives into 47, all the terms cancel and the solution is therefore verified Using 437 and 438 in 439, we can isolate u = Q x to give u = F 1 w x F F 1 w F F 1 F y, 441 which is the noncommutative Miura transformation between the KP and mkp equations

93 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 9 46 Matrix solutions We take trivial the vacuum solution f = 1 so that w = W = 0 This gives ΩP, Θ P f [n+1] = Θ I 44 The eigenfunctions θ i and the binary eigenfunctions ρ i satisfy θ i,xx = θ i,y, θ i,t = 4θ i,xxx and ρ i,xx = ρ i,y, ρ i,t = 4ρ i,xxx The simplest nontrivial solutions of these equations are θ j = A j e η j, ρ i = B i e γ i, where η j = p j x + p j y 4p j t, γ i = q i x + q i y 4q i t and A j, B i are d m matrices With this, we have Ωθ, ρ = δ i,j I p jb T i A j q i p j q i eη j γ i We take A j = r j P j, where r j is a scalar and P j is a projection matrix, and we take B i = I In the case n = 1, expanding 44 gives r q P f [] = I + e Λ rp qp q If r > 0 and either q > p > 0 or 0 > q > p, or alternatively, if r < 0 and either p > q > 0 or 0 > p > q then w = f [],x f 1 [] = 1 4 pq 1 p q P sech W = f [],y f 1 [] = p + qw, Λ + ϕ sech Λ + χ where ϕ = log pr qp q and χ = log r p q Both w and W have a unique maximum where Λ = log pq 1 1 r = ξ p q,

94 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 93 In the case n =, expanding 44 gives f [3] = I + A 1 e η 1 A e η δ i,j I = I + L 1 e γ 1 L e γ = I + 1 q 1 L q L, I e γ 1 q 1 I e γ q 1 p ja j q i p j q i eη j γ i, say I e γ 1 q 1 I e γ q where L 1 and L satisfy Solving for L 1 and L gives where h i = e Λ i L 1 I p 1r 1 e Λ 1 P 1 p 1 q 1 q 1 L I p r e Λ P p q q = e Λ 1 A 1 + p 1e Λ 1 p 1 q q L A 1, = e Λ A + p e Λ p q 1 q 1 L 1 A L 1 = p q 1 q 1 p 1 q q h I + p 1 A A 1, h L = p 1 q q p q 1 q 1 h 1 I + p A 1 A, h p ir i p i q i q i and h = h 1 h q 1 q p 1 q p q 1 αp 1 p r 1 r We now investigate the behaviour of f [3] as t ± without loss of generality that 0 > p > q > p 1 > q 1 Then, as t, and therefore f [3] I + r 1 q 1 P 1 h 1 w 1 4 p 1q 1 1 p1 q 1 Λ1 + ϕ 1 P 1 sech where ϕ 1 = log p1 r 1 q 1 p 1 q 1 ξ1 = log p 1 q r 1 p 1 q 1 We first fix Λ 1 and assume Λ1 + χ 1 sech, 443 and χ 1 = log r1 p 1 q 1 We also have the phase-constant Note that w = f [3],x f 1 [3] and W = f [3],y f 1 [3] are invariant under the transformation

95 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 94 f [3] f [3] C where C is a non-singular constant matrix As t +, we get M f [3] I + h 1 r p q 1 p 1 q p q 1 + αp 1 p r 1 r p q I + p q A I p q A q r p r M I + h 1 r p q 1 p 1 q p q 1 + αp 1 p r 1 r p q I + p q A q r where I + ˆr 1 q 1 ˆP1 e γ 1 η 1 p 1ˆr 1 q 1 p 1 q 1, Therefore M = r p p 1 q p q 1 A 1 p q p 1 p q 1 A A 1 + p p 1 q A 1 A + αp 1 r 1 p q A, ˆr 1 = r 1 1 αp 1 q 1 p q = r 1ˆµ 1, ˆν 1 p 1 q p q 1 µ 1, ν 1, ˆµ 1 = µ 1 p 1p q µ 1, ν µ p p 1 q µ, ν, ˆν 1 = ν 1 q 1p q µ, ν 1 ν q p q 1 µ, ν, ˆP 1 = ˆµ 1 ˆν 1 ˆµ 1, ˆν 1 w 1 4 p 1q 1 1 p1 q 1 ˆP1 sech Λ1 + ϕ + 1 where ϕ + 1 = log p1ˆr 1 q 1 p 1 q 1, χ + 1 = log ˆr1 p 1 q 1 and ξ 1 + = log Similarly, fixing Λ gives w 1 4 p q 1 p q ˆP sech Λ + ϕ w 1 4 p q 1 p q P sech Λ + ϕ + Λ1 + χ + 1 sech, Λ + χ sech Λ + χ + sech p 1 q 1! 1 ˆr 1 p 1 q 1 as as t, t +, where ˆµ = µ p p 1 q 1 µ,ν 1 µ 1 p 1 p q 1 µ 1,ν 1, ˆν = ν q p 1 q 1 µ 1,ν ν 1 q 1 p 1 q µ 1,ν 1, ˆP = ˆµ ˆν ˆµ,ˆν, ϕ = log pˆr q p q, χ = log ˆr p q, ˆr = r 1 αp 1 q 1 p q p 1 q p q 1 = r ˆµ,ˆν µ,ν, ϕ + = log p r q p q and χ + = log r p q The soliton phase-constants are: ξ1 = log pq 1 1 r 1 p 1 q 1, ξ 1 + = log pq 1 1 ˆr 1 p q, ξ = log p q 1 1 ˆr p q and ξ + = log p q 1 1 r p q

96 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 95 The soliton phase shifts i = ξ + i ξ i are 1 = log r1 ˆr 1 ˆr = log β, = log r = log β The characteristics of the two-soliton solution may be summarised in the same way as the matrix solutions of nckp: The matrix amplitude of the first soliton changes from 1 4 p 1q 1 1 p 1 q 1 P 1 to 1 4 p 1q 1 1 p 1 q 1 ˆP1 and the matrix amplitude of the second soliton changes from 1 4 p q 1 p q ˆP to 1 4 p q 1 p q P as t changes from to + If µ 1, ν = 0 P P 1 = 0 or µ, ν 1 = 0 P 1 P = 0 then α = 0 and therefore β = 1, so there is no phase shift but the matrix amplitudes may still change If µ 1, ν = 0 and µ, ν 1 = 0 giving P 1 P = P P 1 = 0 there is no phase shift or change in amplitude and so the solitons have trivial interaction In general, for n 1, expanding 44 gives f [n+1] = I + A 1 e η 1 A e η A n e δ ηn p i,j I A j e η j γ i 1 j p j q i q i = I + L 1 e γ 1 L e γ L n e γn = I + n i=1 I e γ 1 q 1 I e γ q I e γn q n n n Ie γ 1 Ie γ Ie γn 444, say, q i L i 446 In a similar way to that of matrix solutions of nckp, for n > 3, it is very difficult to isolate each L i So we will now only investigate the three-soliton solution When n = 3, 446 gives f [4] = I + 1 q 1 L q L + 1 q 3 L 3 447

97 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 96 From we must have that L 1 = 1 A 1 + p 1L A 1 h 1 q p 1 q + p 1L 3 A 1 q 3 p 1 q 3 L = 1 A + p L 1 A h q 1 p q 1 + p L 3 A q 3 p q 3 L 3 = 1 h 3 A 3 + p 3L 1 A 3 q 1 p 3 q 1 + p 3L A 3 q p 3 q r ip i, 448, 449, 450 where h i = e Λ i p i q i q i, for i = 1,, 3 Solving for L 1, L and L 3 gives h, 3 L 1 = A 1 + p 1L,3 A 1 h1,, 3q q 3 p q 3 p 3 q q p 1 q + p 1L3, A 1, 451 q 3 p 1 q 3 h1, 3 L = A + p L 1,3 A h1,, 3q 1 q 3 p 1 q 3 p 3 q 1 q 1 p q 1 + p L3,1 A, 45 q 3 p q 3 h1, L 3 = A 3 + p 3L 1, A 3 h1,, 3q 1 q p 1 q p q 1 q 1 p 3 q 1 + p 3L,1 A 3, 453 q p 3 q in which hi, j = p i q j p j q i q i q j h i h j p i p j r i r j α i,j, p p 3 r r 3 α,3 h 1 h1,, 3 = q q 3 p q 3 p 3 q p 1 p 3 r 1 r 3 α 1,3 h q 1 q 3 p 1 q 3 p 3 q 1 p 1 p r 1 r α 1, h 3 q 1 q p 1 q p q 1 p 1p p 3 r 1 r r 3 α 1,,3 q 1 q q 3 p q 1 p 1 q 3 p 3 q + α 1,3, p 1 q p q 3 p 3 q 1 + h 1 h h 3, L i,j = q ip j q i hi, j for i, j {1,, 3} and i j h j q j p i q j I + p i A j A i, Substituting into 447 gives in which gives f [4] = I + 1 h1,, 3q 1 q q 3 b,3 A 1 + b 1,3 A + b 1, A 3 + b 1,3, A 1 A + b 1,,3 A 1 A 3 +b,3,1 A A 1 + b,1,3 A A 3 + b 3,,1 A 3 A 1 + b 3,1, A 3 A, 454 hi, j b i,j = p i q j p j q i, p j r j α i,j,k b i,j,k = p k + h jq j p k q j p j q i α i,k p k q i Using the fact that T ra j = r j and deta j = 0, for j = 1,, 3, expanding detω detω = 1 p 1 q 1 κ 1 e Λ 1 p q κ e Λ p 3 q 3 κ 3 e Λ 3 + p 1p q 1 q κ 1 κ β 1, e Λ 1+Λ + p p 3 q q 3 κ κ 3 β,3 e Λ +Λ 3 + p 1p 3 q 1 q 3 κ 1 κ 3 β 1,3 e Λ 1+Λ 3 p 1p p 3 q 1 q q 3 κ 1 κ κ 3 β 1,,3 e Λ 1+Λ +Λ 3 455

98 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 97 If r > 0 and q 3 > p 3 > q > p > q 1 > p 1 > 0 or 0 > q 3 > p 3 > q > p > q 1 > p 1, or alternatively, if r < 0 and p 3 > q 3 > p > q > p 1 > q 1 > 0 or 0 > p 3 > q 3 > p > q > p 1 > q 1 then, β i,j > 0 and κ i > 0 From 455, detω will be positive-definite if α 1,,3 p q 1 p 1 q 3 p 3 q + α 1,3, p 1 q p q 3 p 3 q 1 < 0 The asymptotic forms of each soliton can be determined by following the methods in Chapter 3 It can again be assumed, without loss of generality, that 0 > p 3 > q 3 > p > q > p 1 > q 1 For soliton 1, as t, h i for i =, 3 Then we can see from that L i 0 for i =, 3 The resulting solution may be compared with similar expressions in the two-soliton matrix solution, giving f [4] I + r 1 q1 P 1 e γ 1 η 1 p 1 r 1 q 1 p 1 q 1, where r 1 = r 1 and P 1 = P 1 Therefore w 1 4 p 1q 1 1 p1 q 1 P 1 sech Λ1 + ϕ 1 Λ1 + χ 1 sech, where ϕ 1 = log p 1r1, χ 1 q 1 p 1 q 1 = log r 1 p 1 q 1 The phase-constant of this soliton is ξ 1 = log As t +, h i p ir i q i p i q i p 1 q r 1 p 1 q 1, for i =, 3 Using the fact that w and W are invariant under the transformation f [4] f [4] C, where C is a constant matrix, we have where f [4] I + r + 1 q1 P + 1 e γ 1 η 1 p 1 r + 1 q 1 p 1 q 1 r 1 + = r 1µ + 1, ν+ 1, P 1 + µ 1, ν 1 = µ+ 1 ν+ 1 µ + 1, ν+ 1, µ + 1 = µ 1 + p 1p q p 3 q 3 µ 1, ν 3 µ 3, ν p µ, ν β,3 µ 3, ν 3 p 1 q 3 p 3 q µ 1, ν p 1 q p 3 q 3 + p 1p q 3 p 3 q µ 1, ν µ, ν 3 p 3 µ 3, ν 3 β,3 µ, ν p 1 q p q 3 µ 1, ν 3 p 1 q 3 p q ν 1 + = ν 1 + q 1p q p 3 q 3 q µ, ν β,3 + q 1p q 3 p 3 q q 3 µ 3, ν 3 β,3 µ 3, ν 1 µ, ν 3 µ 3, ν 3 p q 3 p 3 q 1 µ, ν 1, p q 1 p 3 q 3 µ, ν 1 µ 3, ν µ, ν p 3 q p q 1 µ 3, ν 1 p 3 q 1 p q µ µ 3, ν ν 3

99 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 98 So we have where w 1 4 p 1q 1 1 p1 q 1 P + 1 sech Λ1 + ϕ + 1 Λ1 + χ + 1 sech, ϕ + 1 = log p 1r 1 +, χ + 1 q 1 p 1 q 1 = log r+ 1 p 1 q 1 The phase-constant of this soliton is ξ + 1 = log where where p 1 q r + 1 p 1 q 1 Fixing Λ brings soliton to rest As t, h 1 r 1 p 1 q 1 r = r µ, ν µ, ν f [4] I + L 1, q 1 I + + L,1 q r q P e Λ p r q p q = r β 1,, P = µ ν µ, ν,, and h 3 + This gives µ = µ p p 1 q 1 µ, ν 1 p 1 p q 1 µ 1, ν 1 µ 1, ν = ν q p 1 q 1 µ 1, ν q 1 p 1 q µ 1, ν 1 ν 1 As t +, h 1 + and h 3 r 3 p 3 q 3 This gives r + = r µ +, ν+ µ, ν f [4] I + L,3 q 1 I + + L 3, q r q P e Λ p r q p q = r β,3, P + = µ+ ν+ µ +, ν+, µ + = µ p p 3 q 3 µ, ν 3 p 3 p q 3 µ 3, ν 3 µ 3, ν + = ν q p 3 q 3 µ 3, ν q 3 p 3 q µ 3, ν 3 ν 3 So the asymptotic forms for soliton are w 1 4 p q 1 p q P sech Λ + ϕ w 1 4 p q 1 p q P + sech Λ + ϕ + where ϕ = log p r q p q, ϕ + = log p r + q p q Λ + χ sech Λ + χ + sech,, χ = log r p q We also have the soliton phase-constants ξ = log p q 1 1 r, ξ + p q = log p q 1 1 r + p q as as t, t +, and χ + = log r + p q

100 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 99 With Λ 3 fixed, soliton 3 is a rest As t, h i p ir i q i p i q i for i = 1, This gives where r 3 = r 3µ 3, ν 3 µ 3, ν 3 f [4] I + r 3 q3 P 3 e Λ 3 p 3 r 3 q 3 p 3 q 3 = r 3β 1,,3, P3 β = µ 3 ν 3 1, µ 3, ν 3, µ 3 = µ 3 + p 3p q p 1 q 1 µ 3, ν 1 µ 1, ν p µ, ν β 1, µ 1, ν 1 p 3 q 1 p 1 q µ 3, ν p 3 q p 1 q 1 + p 3p q p 1 q 1 µ 3, ν µ, ν 1 p 1 µ 1, ν 1 β 1, µ, ν p 3 q p q 1 µ 3, ν 1 µ 1, p 3 q 1 p q ν3 = ν 3 + q 3p q p 1 q 1 µ 1, ν 3 µ, ν 1 q µ, ν β 1, µ 1, ν 1 p q 1 p 1 q 3 µ, ν 3 p q 3 p 1 q 1 + q 3p q p 1 q 1 µ, ν 3 µ 1, ν q 1 µ 1, ν 1 β 1, µ, ν p 1 q p q 3 µ 1, ν 3 ν 1 p 1 q 3 p q As t +, h i +, for i = 1, This gives, µ ν f [4] I + r + 3 q3 P + 3 e Λ 3 p 3 r + 3 q 3 p 3 q 3 where r + 3 = r 3, P + 3 = µ+ 3 ν+ 3 µ + 3,ν+ 3, µ+ 3 = µ 3 and ν + 3 = ν 3 So the asymptotic forms for soliton 3 are w 1 4 p 3q 3 1 p3 q 3 P 3 sech Λ3 + ϕ 3 w 1 4 p 3q 3 1 p3 q 3 P + 3 sech Λ3 + ϕ + 3 Λ3 + χ 3 sech Λ3 + χ + 3 sech where ϕ 3 = log p3 r 3 q 3 p 3 q 3, ϕ + 3 = log p3 r + 3 q 3 p 3 q 3, χ 3 = log r 3 p 3 q 3 In addition, we have the soliton phase-constants ξ3 = log p 3 q3 1 1 r3, ξ 3 + p 3 q = log p 3 q3 1 1 r p 3 q 3 The soliton phase shifts j = ξ j + ξ j are µ + 1 = log 1, ν 1 + µ 1, ν 1 = log µ + = log, ν + µ, ν = log µ + 3 = log 3, ν 3 + µ 3, ν 3, β1,,3 β,3 β,3 β 1, β1, = log β 1,,3,, as as t, t +, and χ + 3 = log r + 3 p 3 q 3

101 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION Plots of the matrix solutions In this section, we demonstrate the interaction properties of the two- and three-soliton matrix solution of ncmkp with various plots where Figure 41 shows a plot of the generic two-soliton matrix solution w = w ij, i, j = 1,, r 1 = 1 ˆr 1 = 0846, ˆr = 0846 r = 1, µ 1 = 1 0 ˆµ 1 = , µ = 1 05 ˆµ = 15 05, ˆν 1 = ν 1 = 1 1, ˆν = ν = , P 1 = 1 1 ˆP = 1689, ˆP = P = 046, as t changes from to + This plot shows both a change in matrix amplitude and a phase-shift upon interaction where Figure 4 shows a plot of the generic three-soliton matrix solution w = w ij, i, j = 1,,

102 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 101 r1 = 1 r+ 1 = 0711, r = r+ = 1135, r3 = 11 r+ 3 = 1, µ 1 1 = 0333 µ = 005, ν1 1 = ν = 6656, µ 6 = 11 µ = 3315, ν 015 = 5 ν = 4606, µ 3 36 = 038 µ = 0, ν = 1806 ν = 4, P1 = 06 1 P 1 + = , P = P + = , P3 = P 3 + = , as t changes from to + This plot shows both a change in matrix amplitude and a phase-shift upon interaction

103 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 10 Figure 41: Plot of matrix mkp two-soliton solution at t = 0 with parameters p 1 = 1 4, p = 1 4, q 1 = 3 4, q = 3 4, r 1 = 1 and r = 1

104 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 103 Figure 4: Plot of matrix mkp three-soliton solution at t = 0 with parameters p 1 = 1, p = 5, p 3 = 6, q 1 =, q = 5, q 3 = 9 and r 1 = r = r 3 = 1 48 The noncommutative Miura transformation Using the same method as in the commutative case, we can construct a noncommutative Miura transformation between the nckp and ncmkp equations Upon calculation of the transformed pseudodifferential operator L KP = θ 1 L KP θ, we have L KP = + θ 1 θ x + 1 θ 1 uθ 1 + θ 1 u θ 1 θ 1 uθ x + Comparing this with the operator L mkp = x + w + w 1 1 x + w x + w 3 3 x + 456

105 CHAPTER 4 A NONCOMMUTATIVE MKP EQUATION 104 and equating coefficients gives w = θ 1 θ x, 457 w 1 = 1 θ 1 uθ, 458 w = θ 1 u θ 1 θ 1 uθ x 459 These coefficients will satisfy 43, 44 and 45 Isolating u in 458 gives u = θw 1 θ Given that w = θ 1 θ x and w = f x f 1, we can conclude that f = θ 1 Therefore, we can rewrite 460 as u = f 1 w 1 f, 461 which was also derived in [8] Upon substitution of 410 in 461 we obtain the noncommutative Miura transformation between the nckp and ncmkp equations: u = f 1 w x f f 1 w f f 1 f y 46 = f 1 f xx f x f 1 f x f 1 f 1 f y 463 Direct substitution of 463 into nckp 319 leads to the left-hand side of 319 being identically zero Therefore, 463 defines a new solution of nckp 319 Furthermore, 46 is consistent with the nc Miura transformation 441

106 Chapter 5 Dromions of the matrix equations The aim of this chapter is to find exponentially localized structures, obtained from the matrix versions of the nc KP and nc mkp equations The commutative KP equation [3] and the commutative Davey-Stewartson DS equations [49] are known to have localized lump solutions, which have algebraic decay at infinity However, when each lump collides, the interaction is completely trivial Equations such as the DSI equations [, 18] and the Nizhnik-Veselov-Novikov NVN [4] equations are known to have localized solutions which have exponential decay at infinity In this case, the solutions have interesting interaction properties such as changes in amplitude and trajectory For the NVN equations, it was shown, in [50], that an exponentially localized solution may be thought of as a two-soliton solution made out of two intersecting ghost solitons In [18], the authors show, by means of direct methods, how to derive the characteristics of exponentially localized solutions of the DSI equations This has been extended to the noncommutative setting in [17] In both the DSI and NVN equations, the underlying solitons which interact to create the localized solution are perpendicular These localized solutions are called dromions which comes from the Greek word dromos meaning track This term was coined in [11] because the dromions are located at soliton interactions which can be thought of as forming tracks 51 Matrix KP single dromion Since the one-soliton matrix solutions of nckp and ncmkp have projection matrices governing their amplitude, their determinant will equal zero Therefore, the natural thing to investigate when looking for dromions of these solutions, is the determinant of the two-soltion matrix solution We begin by investigating the two-soliton matrix solution of 105

107 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 106 nckp Recall from Chapter 3 that the two-soliton matrix solution of nckp is u = v x where in which and K 1 = p q 1 h K = p 1 q h v = K 1 + K, h = h 1 h p 1 q p q 1 αr 1 r, h i = e Λ i + h p 1 q I A A 1, h 1 p q 1 I A 1 A, r i p i q i and α = µ 1, ν µ, ν 1 µ 1, ν 1 µ, ν = T rp 1P Since the determinant of a projection matrix is zero and the trace of a projection matrix is equal to its rank, detu can easily be expanded along any row or column In doing so, the result greatly simplifies and we obtain detu = 4r 1 r p 1 q p q 1 1 α h1 h p 1 q p q 1 h h x x r 1 r α 1 h = 4r 1 r p 1 q p q 1 1 αh 1,x h,x h 51 The single dromion 51 can be rewritten as detu = 4r 1r p 1 q 1 p q 1 αe Λ 1+Λ e Λ 1 +Λ + κ 1 e Λ + κ e Λ 1 + κ 5 = 4r 1 r p 1 q 1 p q 1 α e 1 Λ 1+Λ + κ 1 e 1 Λ 1 Λ + κ e 1 Λ Λ 1 + κe 1 Λ 1+Λ 53 x where κ i = r i p i q i, for i = 1, and κ = κ 1κ β, where β = 1 αp 1 q 1 p q p 1 q p q 1 Figure 5 shows a plot of the dromion 53 The method of describing the characteristics of this dromion is in the spirit of that in [18] and [50] with one main difference being that the solitons governing the dromion are not necessarily perpendicular to one another The characteristics of detu as given by equation 53 may be summarised by the following theorem: Theorem If detω is positive-definite and if α 1, then detu has the following properties:

108 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS detu decays to zero exponentially as x, y in any direction and has a unique maximum or minimum value detu max/min = 1 αp 1 q 1 p q β The dromion will have negative, zero or positive amplitude The amplitude is negative if α > 1, zero if α = 1, positive if α < 1 At time t this maximum or minimum is located at x, y = 1 l 1, l ξ 1 + ξ+ 1 l 1 ξ + ξ+ + 8l,3t, where l i,j = l i i l j j l j i l i j and l j i l 1 1 ξ + ξ+ l1 ξ 1 + ξ l 1,3t, 55 = q j i pj i This result implies that the dromion is located symmetrically between the solitons in the two-soliton matrix solution as illustrated by Figure 51 3 The trajectory of the dromion is the straight line l 1 1 l,3 + l 1 l 1,3 y = l1,3 l,3 x + ξ1 + ξ l 1, l,3 ξ + ξ+ l 1 l,3 + l l 1,3 soliton phase-shift dromion soliton phase-shift Figure 51: Phase-shifts in the solitons and the location of the dromion

109 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 108 Proof From 53, we see that detu decays to zero exponentially as x, y in any direction since, along any ray in the x, y-plane, at least one of the exponentials in the denominator is unbounded as x, y approaches infinity To see this, let y = kx, where k R, be a ray in any direction Substituting this into 53 gives e 1 +κ e 1 l 1 1 +l1 + l 1 +l k x 4 l 3 1 +l3 t l 1 l1 1 + l l 1 k x 4 l 3 l3 1 t + κ 1 e 1 + κe 1 l 1 1 l1 + l 1 l k x 4 l 3 1 l3 t l 1 1 +l1 + l 1 +l k x 4 l 3 1 +l3 t on the denominator This expression must tend to infinity for any values of k and l j i, i, j = 1, as x ± Since detu is exponentially localized, a unique critical point must be either a maximum or a minimum If we consider the conditions that α 1 and detu x and detu y vanish simultaneously, we get l l 1 e 1 Λ 1+Λ + κ 1 +κ l 1 1 l 1 e 1 Λ Λ 1 + κ l l 1 l 1 1 l 1 e 1 Λ 1 Λ e 1 Λ 1+Λ = 0 and l 1 + l e 1 Λ 1+Λ + κ 1 +κ l 1 l e 1 Λ Λ 1 + κ l 1 + l l 1 l e 1 Λ 1 Λ e 1 Λ 1+Λ = 0 which imply that e Λ 1+Λ = κ and e Λ 1+Λ = κ 1 κ and so e Λ 1 = κ1 κ = κ 1 β and e Λ κ κ = = κ β 57 κ κ 1 Substituting 57 into 5 gives the maximum or minimum of detu Solving 57 for x and y gives 55, the location of the dromion Eliminating t in 55 gives the trajectory of the dromion The dromion as given by 53 is still prevalent when there is no phase-shift and when there is both no phase-shift and no change in amplitude This is different from the DSI and NVN equations where the dromion vanishes when there is no phase-shift

110 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 109 a b Figure 5: a Plot of a single dromion at t = 0 with parameters r 1 =, r = 1, p 1 = 1 4, q 1 = 3 4, p = 3 4, q = 1 4, µt 1 = 1, µt = , ν T 1 = and ν T = 1 3 b Plot of the corresponding nckp two-soliton matrix solution

111 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS A three-dromion example The determinant of the three-soliton matrix solution gives a three-dromion structure In this case, the three dromions will always collide as they are effectively sharing the same origin This is illustrated in Figure 53 Adding more solitons to the solution would give a much more complicated dromion scattering scheme in which all dromions may not simultaneously collide Therefore, we will concentrate on the three-dromion case here and perform a detailed asymptotic analysis Figure 53: Schematic form of the dromion scattering Upon expansion of detu = detv [4],x, where u is the three-soliton matrix solution of nckp, by using the fact that the trace of a projection matrix is equal to its rank and its determinant is equal to zero we get detu = h 1,xh,x m 1,,3 + h 1,x h 3,x m 1,3, + h,x h 3,x m,3,1 h, 58 1,, 3 where we have the quadratic equations in h k : m i,j,k = r i r j h k 1 α αi,j,k l i,k + l k,j i,j + h k r k α i,k,jl j,k + l k,i l i,k l k,j α i,kl i,k + l k,i l i,k l k,i + α j,k l j,k l k,j + α i,k l i,k l k,i rk, α j,kl j,k + l k,j l j,k l k,i l j,k l k,j αi,j,k + α i,k,j + α i,kα j,k l j,k l k,i l i,k l k,j l i,j l k,j l j,k l k,i l i,k l k,i l j,k l k,j in which l i,j = p i q j for i, j, k {1,, 3} and i j k To investigate the behaviour of each dromion as t ±, we fix attention on the dromion arising from the interaction of the ith and jth solitons, which we term di, j In addition, the corresponding two-soliton interaction matrix potential will be termed v i,j We consider detu as given by 58 in a

112 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 111 frame moving with the i, jth dromion by rewriting it in terms of q 3 j p 3 j x = ˆx + 4 q i p i q3 i p3 i q j p j q j p j qi p i q j p j q t, i p i q 3 i p 3 i y = ŷ + 4 q j p j qj 3 p3 j q i p i q j p j qi p i q j p j q t, i p i from which we obtain Λ i = q i p i ˆx + q i p i ŷ, Λ j = q j p j ˆx + q j p jŷ, for i, j = 1,, 3 In accordance with the three-soliton matrix solution, we will assume, without loss of generality, that 0 > p 3 > q 3 > p > q > p 1 > q 1 Let us begin with the frame moving with d1, To obtain the characteristics of this dromion, we let soliton 3 pass through solitons 1 and, which are stationary, as t ± We then study the asymptotic behaviour of the resulting two-soliton interaction With solitons 1 and fixed, h 1 and h are are also fixed and we study the asymptotic behaviour of h 3 as t ± We have that r 3 p h 3 3 q 3 as t, + as t + When h 3 r 3 /p 3 q 3, equation 351 gives where v 1, = p 3 q 3 b 1, A 3 + b 1,,3 A 1 A 3 + b 3,,1 A 3 A 1 + b,1,3 A A 3 + b 3,1, A 3 A h1, r 3 r3 h r r 3 α,3 r3 h 1 r 1 r 3 α 1,3 + A 1 + A p 3 q 3 p q 3 p 3 q p 3 q 3 p 1 q 3 p 3 q 1 r 3 α 1,3, r 3 + A 1 A p q 3 p 3 q 1 α 1, p q 1 p 3 q 3 r 3 α 1,,3 r 3 + A A 1, 59 p 1 q 3 p 3 q α 1, p 1 q p 3 q 3 h1, = h 1 h r α,3 p 3 q 3 h 1 p q 3 p 3 q r 1α 1,3 p 3 q 3 h p 1 q 3 p 3 q 1 r 1 r α 1, p 1 q p q 1 +r 1 r p 3 q 3 α 1,,3 p q 1 p 1 q 3 p 3 q + α 1,3, p 1 q p q 3 p 3 q 1 To obtain the characteristics of the dromion d1,, we find the asymptotic forms of ũ := v 1, x Firstly, let us fix Λ 1 Since ũ is invariant under the transformation v 1, v 1, +C,

113 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 11 where C is a constant matrix, we have where v 1, = r 11, P 1 1, e Λ 1 + r 1 1, p 1 q 1 as t, v + 1, = r 11, + P 1 1, + e Λ 1 + r 1 1, + p 1 q 1 as t +, P 1 1, = µ 11, ν 1 1, T µ 1 1,, ν 1 1,, P 11, + = µ 11, + ν 1 1, +T µ 1 1, +, ν 1 1, +, µ 1 1, = µ 1 p 3 q 3 µ 1, ν 3 µ 3 p 1 q 3 µ 3, ν 3, ν 11, = ν 1 p 3 q 3 µ 3, ν 1 ν 3 p 3 q 1 µ 3, ν 3, µ 1 1, + = µ 1 + p q p 3 q 3 µ 1, ν 3 µ 3, ν µ, ν β,3 µ 3, ν 3 p 1 q 3 p 3 q µ 1, ν p 1 q p 3 q 3 + p q 3 p 3 q µ 1, ν µ, ν 3 µ 3, ν 3 β,3 µ, ν p 1 q p q 3 µ 1, ν 3 µ 3, p 1 q 3 p q ν 1 1, + = ν 1 + p q p 3 q 3 µ 3, ν 1 µ, ν 3 µ, ν β,3 µ 3, ν 3 p q 3 p 3 q 1 µ, ν 1 p q 1 p 3 q 3 + p q 3 p 3 q µ, ν 1 µ 3, ν µ 3, ν 3 β,3 µ, ν p 3 q p q 1 µ 3, ν 1 ν 3, p 3 q 1 p q r 1 1, = r 1µ 1 1,, ν 1 1, µ 1, ν 1 r 1 1, + = r 1µ 1 1, +, ν 1 1, + µ 1, ν 1 = r 1 β 1,3 and = r 1 β 1,,3 β,3 µ ν These asymptotic expressions for v 1, are of the same form as the one-soliton matrix potential v [] discussed in Chapter 3 Therefore, the asymptotic forms for ũ are ũ 1 1 p 1 q 1 P 1 1, sech Λ 1 + ξ 1 1, ũ 1 1 p 1 q 1 P 1 1, + sech Λ 1 + ξ 1 1, + as as t, t +, with phase-constants ξ 1 1, = log r1 1, p 1 q 1 and ξ 1 1, + = log r1 1, + p 1 q 1 Next we fix Λ Since ũ is invariant under the transformation v 1, v 1, + C, where C is a constant matrix, we have v 1, = r 1, P 1, e Λ + r 1, p q as t, v + 1, = r 1, + P 1, + e Λ + r 1, + p q as t +,

114 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 113 in which P 1, = µ 1, ν 1, T µ 1,, ν 1, P 1, + = µ 1, + ν 1, +T µ 1, +, ν 1, +, µ 1, + = µ p 3 q 3 µ, ν 3 µ 3 p q 3 µ 3, ν 3, ν 1, + = ν p 3 q 3 µ 3, ν ν 3 p 3 q µ 3, ν 3, µ 1, = µ + p 1 q 1 p 3 q 3 µ, ν 3 µ 3, ν 1 µ 1, ν 1 β 1,3 µ 3, ν 3 p q 3 p 3 q 1 µ, ν 1 p q 1 p 3 q 3 + p 1 q 1 p 3 q 3 µ, ν 1 µ 1, ν 3 µ 3, ν 3 β 1,3 µ 1, ν 1 p q 1 p 1 q 3 µ, ν 3 µ 3, p q 3 p 1 q 1 ν 1, = ν + p 1 q 1 p 3 q 3 µ 3, ν µ 1, ν 3 µ 1, ν 1 β 1,3 µ 3, ν 3 p 1 q 3 p 3 q µ 1, ν p 1 q p 3 q 3 + p 1 q 1 p 3 q 3 µ 1, ν µ 3, ν 1 µ 3, ν 3 β 1,3 µ 1, ν 1 p 3 q 1 p 1 q µ 3, ν ν 3, p 3 q p 1 q 1 r 1, = r µ 1,, ν 1, µ, ν r 1, + = r µ 1, +, ν 1, + µ, ν = r β 1,,3 β 1,3 = r β,3 and µ 1 ν 1 So the asymptotic forms for ũ are ũ 1 1 p q P 1, sech Λ + ξ 1, ũ 1 1 p q P 1, + sech Λ + ξ 1, + as as t, t + The phase-constants are: ξ 1, = log r 1, p q and ξ 1, + = log r 1, + p q Furthermore, the soliton phase-shifts j 1, = ξ j 1, + ξ j 1,, for j = 1, are 1 1, = log β1, and 1, = log β1,, in which β1, = β 1,,3 β 1,3 β,3 The asymptotic expressions for ũ can now be used to describe the dromion d1, as t When h 3 r 3 /p 3 q 3, equation 58 gives d1, 4r 1 1, r 1, + p 1 q 1 p q 1 α 1, e 1 Λ 1+Λ + κ 1 1, e 1 Λ 1 Λ + κ 1, e 1 Λ Λ 1 + κ 1, e 1 Λ 1+Λ, 510 where α 1, = T rp 11, P 1, +, κ 1, = κ 1 κ β 1, κ 1 1, = r 11,, κ 1, = r 1, + p 1 q 1 p q and In Chapter 3, we had that K i,j = p j q i hi, j h jp i q j I A j A i,

115 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 114 for i, j {1,, 3} and i j When h 3 +, from 347 we have that K 3 0 and therefore v 1, K 1, + K,1 511 The asymptotic expression 511 is of the same form as the two-soliton matrix potential v [3] and the resulting dromion is therefore of the same form as the single dromion as given by 53 Therefore, when Λ 1 is fixed, the asymptotic forms for ˆũ := v 1, x are ˆũ 1 1 p 1 q 1 ˆP1 1, sech Λ 1 + ˆξ 1 1, ˆũ 1 1 p 1 q 1 ˆP1 1, + sech Λ 1 + ˆξ 1 1, + as as t, t +, where ˆP 1 1, = ˆµ 11, ˆν 1 1, T ˆµ 1 1,, ˆν 1 1,, ˆP1 1, + = ˆµ 11, +ˆν 1 1, +T ˆµ 1 1, +, ˆν 1 1, +, ˆµ 1 1, = µ 1, ˆν 1 1, = ν 1, ˆµ 1 1, + = µ 1 p q µ 1, ν µ p 1 q µ, ν, ˆν 1 1, + = ν 1 p q µ, ν 1 ν p q 1 µ, ν, ˆr 11, = r 1, ˆr 1 1, + = r 1ˆµ 1, ˆν 1 µ 1, ν 1 ˆξ 1 1, ˆr1 1, = log and p 1 q ˆξ 1 1, + ˆr1 1, + = log 1 p 1 q 1 = r 1 β 1,, When Λ is fixed, the asymptotic forms for ˆũ are ˆũ 1 1 p q ˆP 1, sech Λ + ˆξ 1, ˆũ 1 1 p q ˆP 1, + sech Λ + ˆξ 1, + as as t, t +, in which ˆP 1, = ˆµ 1, ˆν 1, T ˆµ 1,, ˆν 1,, ˆP 1, + = ˆµ 1, +ˆν 1, +T ˆµ 1, +, ˆν 1, +, ˆµ 1, + = µ, ˆν 1, + = ν, ˆµ 1, = µ p 1 q 1 µ, ν 1 µ 1 p q 1 µ 1, ν 1, ˆν 1, = ν p 1 q 1 µ 1, ν ν 1 p 1 q µ 1, ν 1, ˆr 1, = r ˆµ, ˆ ν = r β 1,, ˆr 1, + = r, µ, ν ˆr ˆξ 1, = log and p q ˆξ 1, + ˆr 1, + = log p q The soliton phase-shifts ˆ j 1, = ˆξ j 1, + ˆξ j 1,, for j = 1, are ˆ 1 1, = log β 1, + and ˆ 1, = log β 1, +, where β 1, + = β 1,

116 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 115 d1, The dromion d1, as t + can now be written as 4ˆr 1 1, ˆr 1, + p 1 q 1 p q 1 α + 1, e 1 Λ 1+Λ + κ 1 1, + e 1 Λ 1 Λ + κ 1, + e 1 Λ Λ 1 + κ + 1, e 1 Λ 1+Λ, 51 where α + 1, = T r ˆP 1 1, ˆP 1, +, κ 1 1, + = κ 1, κ 1, + = κ and κ + 1, = κ 11, + κ 1, + β + 1, When in a frame moving with the dromion d, 3, the asymptotic analysis is very similar to that of d1, In this case, h and h 3 are fixed and r 1 p h 1 1 q 1 as t, + as t + When h 1 r 1 p 1 q 1, fixing Λ gives the asymptotic forms for ũ: ũ 1 1 p q P, 3 sech Λ + ξ, 3 as ũ 1 1 p q P, 3 + sech Λ + ξ, 3 + as t, t +, with phase-constants ξ, 3 = log r,3 p q and ξ, 3 + = log r,3 + p q and P, 3 = µ, 3 ν, 3 T µ, 3, ν, 3, P, 3 + = µ, 3 + ν, 3 +T µ, 3 +, ν, 3 +, µ, 3 = µ p 1 q 1 µ, ν 1 µ 1 p q 1 µ 1, ν 1, ν, 3 = ν p 1 q 1 µ 1, ν ν 1 p 1 q µ 1, ν 1, µ, 3 + = µ + p 1 q 1 p 3 q 3 µ, ν 3 µ 3, ν 1 µ 1, ν 1 β 1,3 µ 3, ν 3 p q 3 p 3 q 1 µ, ν 1 p q 1 p 3 q 3 + p 1 q 1 p 3 q 3 µ, ν 1 µ 1, ν 3 µ 3, ν 3 β 1,3 µ 1, ν 1 p q 1 p 1 q 3 µ, ν 3 µ 3, p q 3 p 1 q 1 ν, 3 + = ν + p 1 q 1 p 3 q 3 µ 3, ν µ 1, ν 3 µ 1, ν 1 β 1,3 µ 3, ν 3 p 1 q 3 p 3 q µ 1, ν p 1 q p 3 q 3 + p 1 q 1 p 3 q 3 µ 1, ν µ 3, ν 1 µ 3, ν 3 β 1,3 µ 1, ν 1 p 3 q 1 p 1 q µ 3, ν ν 3, p 3 q p 1 q 1 r, 3 = r µ, 3, ν, 3 µ, ν r, 3 + = r µ, 3 +, ν, 3 + µ, ν = r β 1, and = r β 1,,3 β 1,3 µ 1 ν 1 Fixing Λ 3 gives the asymptotic forms for ũ: ũ 1 1 p 3 q 3 P 3, 3 sech Λ 3 + ξ 3, 3 ũ 1 1 p 3 q 3 P 3, 3 + sech Λ 3 + ξ 3, 3 + as as t, t +,

117 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 116 where ξ 3, 3 = log r3,3 p 3 q 3 and ξ 3, 3 + = log r3,3 + p 3 q 3 and P 3, 3 = µ 3, 3 ν 3, 3 T µ 3, ν 3, P 3, 3 + = µ 3, 3 + ν 3, 3 +T µ 3, 3 +, ν 3, 3 +, µ 3, 3 + = µ p 3 q 3 µ, ν 3 µ 3 p q 3 µ 3, ν 3, ν 3, 3 + = µ p 3 q 3 µ, ν 3 µ 3 p q 3 µ 3, ν 3, µ 3, 3 = µ 3 + p q p 1 q 1 µ 3, ν 1 µ 1, ν µ, ν β 1, µ 1, ν 1 p 3 q 1 p 1 q µ 3, ν p 3 q p 1 q 1 + p q p 1 q 1 µ 3, ν µ, ν 1 µ 1, ν 1 β 1, µ, ν p 3 q p q 1 µ 3, ν 1 µ 1, p 3 q 1 p q ν 3 = ν 3 + p q p 1 q 1 µ 1, ν 3 µ, ν 1 µ, ν β 1, µ 1, ν 1 p q 1 p 1 q 3 µ, ν 3 p q 3 p 1 q 1 + p q p 1 q 1 µ, ν 3 µ 1, ν µ 1, ν 1 β 1, µ, ν p 1 q p q 3 µ 1, ν 3 ν 1, p 1 q 3 p q r 3, 3 = r 3µ 3, 3, ν 3, 3 µ 3, ν 3 r 3, 3 + = r 3µ 3, 3 +, ν 3, 3 + µ 3, ν 3 = r 3 β 1,,3 β 1, = r 3 β,3 and The soliton phase-shifts j, 3 = ξ j, 3 + ξ j, 3, for j =, 3 are, 3 = log β,3 and 3, 3 = log β,3, in which β,3 = β 1,,3 β 1, β 1,3 µ ν The asymptotic expressions for ũ can be used to describe the dromion d, 3 as t When h 1 r 1 /p 1 q 1, equation 58 gives d, 3 4r, 3 r 3, 3 + p q p 3 q 3 1 α,3 e 1 Λ +Λ 3 + κ, 3 e 1 Λ Λ 3 + κ 3, 3 e 1 Λ 3 Λ + κ,3 e 1 Λ +Λ 3, 513 where α,3 = T rp, 3 P 3, 3 +, κ, 3 = r, 3 p q, κ 3, 3 = r 3, 3 p 3 q 3, and κ,3 = κ, 3 κ 3, 3 β,3 In the case that h 1 +, when Λ is fixed, the asymptotic forms for ˆũ := v,3 x are ˆũ 1 1 p q ˆP, 3 sech Λ + ˆξ, 3 ˆũ 1 1 p q ˆP, 3 + sech Λ + ˆξ, 3 + as as t, t +,

118 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 117 in which ˆP, 3 = ˆµ, 3 ˆν, 3 T ˆµ, 3, ˆν, 3, ˆP, 3 + = ˆµ, 3 +ˆν, 3 +T ˆµ, 3 +, ˆν, 3 +, ˆµ, 3 = µ, ˆν, 3 = ν, ˆµ, 3 + = µ p 3 q 3 µ, ν 3 µ 3 p q 3 µ 3, ν 3 ˆν, 3 + = ν p 3 q 3 µ 3, ν ν 3 ˆr, 3 = r, p 3 q µ 3, ν 3 ˆr, 3 + = r ˆµ, 3, ˆν, 3 = r β,3, ˆξ, 3 ˆr, 3 = log µ, ν p q ˆξ, 3 + ˆr, 3 + = log p q and When Λ 3 is fixed, the asymptotic forms for ˆũ are ˆũ 1 1 p 3 q 3 ˆP3, 3 sech Λ 3 + ˆξ 3, 3 ˆũ 1 1 p 3 q 3 ˆP3, 3 + sech Λ 3 + ˆξ 3, 3 + as as t, t +, where ˆP 3, 3 = ˆµ 3, 3 ˆν 3, 3 T ˆµ 3, 3, ˆν 3, 3, ˆP3, 3 + = ˆµ 3, 3 +ˆν 3, 3 +T ˆµ 3, 3 +, ˆν 3, 3 +, ˆµ 3, 3 + = µ 3, ˆν 3, 3 + = ν 3, ˆµ 3, 3 = µ 3 p q µ 3, ν µ p 3 q µ, ν, ˆν 3, 3 = ν 3 p q µ, ν 3 ν p q 3 µ, ν, ˆr 3, 3 = r 3ˆµ 3, 3, ˆν 3, 3 = r 3 β,3, µ 3, ν 3 ˆr 3, 3 + = r 3, ˆξ3, 3 ˆr3, 3 = log and p 3 q 3 ˆξ 3, 3 + ˆr3, 3 + = log p 3 q 3 Furthermore, the soliton phase-shifts are ˆ j, 3 = ˆξ j, 3 + ˆξ j, 3, for j =, 3 are d, 3 ˆ = log β,3 + and ˆ 3 = log β,3 +, where β,3 + = β,3 The dromion d, 3 as t + can now be written as 4ˆr, 3 ˆr 3, 3 + p q p 3 q 3 1 α +,3 e 1 Λ +Λ 3 + κ, 3 + e 1 Λ Λ 3 + κ 3, 3 + e 1 Λ 3 Λ + κ +,3 e 1 Λ +Λ 3, 514 where T r ˆP, 3 ˆP3, 3 +, κ, 3 + = κ, κ 3, 3 + = κ 3 and κ +,3 = κ, 3 + κ 3, 3 + β +,3 Finally, we move to a frame moving with d1, 3 In this case h 1 and h 3 are fixed and + as t, h r p q as t +

119 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 118 When h +, fixing Λ 1 gives the asymptotic forms for ũ := v 1,3 x : ũ 1 1 p 1 q 1 P 1 1, 3 sech Λ 1 + ξ 1 1, 3 ũ 1 1 p 1 q 1 P 1 1, 3 + sech Λ 1 + ξ 1 1, 3 + as as t, t +, where P 1 1, 3 = µ 11, 3 ν 1 1, 3 T µ 1 1, 3, ν 1 1, 3, P 11, 3 + = µ 11, 3 + ν 1 1, 3 +T µ 1 1, 3 +, ν 1 1, 3 +, µ 1 1, 3 = µ 1, ν 1 1, 3 = ν 1, µ 1 1, 3 + = µ 1 p 3 q 3 µ 1, ν 3 µ 3 p 1 q 3 µ 3, ν 3 ν 1 1, 3 + = ν 1 p 3 q 3 µ 3, ν 1 ν 3 p 3 q 1 µ 3, ν 3 ξ 1 1, 3 r1 1, 3 = log and p 1 q 1 r 1 1, 3 = r 1, r 1 1, 3 + = r 1µ 1, ν 1 µ 1, ν 1 ξ 1 1, 3 + r1 1, 3 + = log p 1 q 1 = r 1 β 1,3, When Λ 3 is fixed, the asymptotic forms for ũ are ũ 1 1 p 3 q 3 P 3 1, 3 sech Λ 3 + ξ 3 1, 3 ũ 1 1 p 3 q 3 P 3 1, 3 + sech Λ 3 + ξ 3 1, 3 + as as t, t +, where P 3 1, 3 = µ 31, 3 ν 3 1, 3 T µ 3 1, 3, ν 3 1, 3, P 31, 3 + = µ 31, 3 + ν 3 1, 3 +T µ 3 1, 3 +, ν 3 1, 3 +, µ 3 1, 3 + = µ 3, ν 3 1, 3 + = ν 3, µ 3 1, 3 = µ 3 p 1 q 1 µ 3, ν 1 µ 1 p 3 q 1 µ 1, ν 1, ν 3 1, 3 = ν 3 p 1 q 1 µ 1, ν 3 ν 1 p 1 q 3 µ 1, ν 1, r 31, 3 = r 3µ 3 1, 3, ν 3 1, 3 = r 3 β 1,3, µ 3, ν 3 r 3 1, 3 + = r 3, ξ 3 1, 3 r3 1, 3 = log and p 3 q 3 ξ 3 1, 3 + r3 1, 3 + = log p 3 q 3 In addition, we have the soliton phase-shifts j 1, 3 = ξ j 1, 3 + ξ j 1, 3, for j = 1, 3, which are 1 1, 3 = log β1,3 and 3 1, 3 = log β1,3, in which β1,3 = β 1,3 d1, 3 The dromion d1, 3 as t can now be written as 4r 1 1, 3 r 3 1, 3 + p 1 q 1 p 3 q 3 1 α 1,3 e 1 Λ 1+Λ 3 + κ 1 1, 3 e 1 Λ 1 Λ 3 + κ 3 1, 3 e 1 Λ 3 Λ 1 + κ 1,3 e 1 Λ 1+Λ 3, 515

120 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 119 where α 1,3 = T rp 11, 3 P 3 1, 3 +, κ 1 1, 3 = κ 1, κ 3 1, 3 = κ 3, κ 1,3 = κ 11, 3 κ 3 1, 3 β 1,3 and β 1,3 = β 1,3 When h 1 r 1 /p 1 q 1, fixing Λ 1 gives the asymptotic forms for ˆũ: ˆũ 1 1 p 1 q 1 ˆP1 1, 3 sech Λ 1 + ˆξ 1 1, 3 ˆũ 1 1 p 1 q 1 ˆP1 1, 3 + sech Λ 1 + ˆξ 1 1, 3 + as as t, t +, with phase-constants ˆξ 1 1, 3 = log ˆr1 1,3 p 1 q 1 and ˆξ 1 1, 3 + = log ˆr1 1,3 + p 1 q 1 and ˆP 1 1, 3 = ˆµ 11, 3 ˆν 1 1, 3 T ˆµ 1 1, 3, ˆν 1 1, 3, ˆP1 1, 3 + = ˆµ 11, 3 +ˆν 1 1, 3 +T ˆµ 1 1, 3 +, ˆν 1 1, 3 +, ˆµ 1 1, 3 = µ 1 p q µ 1, ν µ p 1 q µ, ν, ˆν 11, 3 = ν 1 p q µ, ν 1 ν p q 1 µ, ν, ˆµ 1 1, 3 + = µ 1 + p q p 3 q 3 µ 1, ν 3 µ 3, ν µ, ν β,3 µ 3, ν 3 p 1 q 3 p 3 q µ 1, ν p 1 q p 3 q 3 + p q 3 p 3 q µ 1, ν µ, ν 3 µ 3, ν 3 β,3 µ, ν p 1 q p q 3 µ 1, ν 3 µ 3, p 1 q 3 p q ˆν 1 1, 3 + = ν 1 + p q p 3 q 3 µ 3, ν 1 µ, ν 3 µ, ν β,3 µ 3, ν 3 p q 3 p 3 q 1 µ, ν 1 p q 1 p 3 q 3 + p q 3 p 3 q µ, ν 1 µ 3, ν µ 3, ν 3 β,3 µ, ν p 3 q p q 1 µ 3, ν 1 ν 3, p 3 q 1 p q ˆr 1 1, 3 = r 1ˆµ 1, ˆν 1 µ 1, ν 1 = r 1 β 1, and ˆ r + 1 = r 1ˆµ 1 1, 3 +, ˆν 1 1, 3 + µ 1, ν 1 = r 1 β 1,,3 β,3 µ ν Next we fix Λ 3 Then the asymptotic forms for ˆũ are ˆũ 1 1 p 3 q 3 ˆP3 1, 3 sech Λ 3 + ˆξ 3 1, 3 ˆũ 1 1 p 3 q 3 ˆP3 1, 3 + sech Λ 3 + ˆξ 3 1, 3 + as as t, t +

121 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 10 The phase-constants are: ˆξ 3 = log ˆr 3 p 3 q 3 and ˆξ 3 + = log ˆr + 3 p 3 q 3 and we also have ˆP 3 1, 3 = ˆµ 31, 3 ˆν 3 1, 3 T ˆµ 3 1, 3, ˆν 3 1, 3, ˆP3 1, 3 + = ˆµ 31, 3 +ˆν 3 1, 3 +T ˆµ 3 1, 3 +, ˆν 3 1, 3 +, ˆµ 3 1, 3 + = µ 3 p q µ 3, ν µ p 3 q µ, ν, ˆν 31, 3 + = ν 3 p q µ, ν 3 ν 3 p q 3 µ, ν, ˆµ 3 1, 3 = µ 3 + p q p 1 q 1 µ 3, ν 1 µ 1, ν µ, ν β 1, µ 1, ν 1 p 3 q 1 p 1 q µ 3, ν p 3 q p 1 q 1 + p q p 1 q 1 µ 3, ν µ, ν 1 µ 1, ν 1 β 1, µ, ν p 3 q p q 1 µ 3, ν 1 µ 1, p 3 q 1 p q ˆν 3 1, 3 = ν 3 + p q p 1 q 1 µ 1, ν 3 µ, ν 1 µ, ν β 1, µ 1, ν 1 p q 1 p 1 q 3 µ, ν 3 p q 3 p 1 q 1 + p q p 1 q 1 µ, ν 3 µ 1, ν µ 1, ν 1 β 1, µ, ν p 1 q p q 3 µ 1, ν 3 ν 1, p 1 q 3 p q ˆr 3 1, 3 = r 3ˆµ 3 1, 3, ˆν 3 1, 3 µ 3, ν 3 ˆr 3 1, 3 + = r 3ˆµ 3 1, 3 +, ˆν 3 1, 3 + µ 3, ν 3 = r 3β 1,,3 β 1, = r 3 β,3 Furthermore, the soliton phase-shifts ˆ j 1, 3 = ˆξ j 1, 3 + ˆξ j 1, 3, for j = 1, 3 are ˆ 1 1, 3 = log β 1,3 + and ˆ 3 1, 3 = log β 1,3 +, where β 1,3 + = β 1,,3 β 1, β,3 and µ ν The asymptotic expressions for ˆũ can again be used to describe the dromion d1, 3 as t + When h r /p q, equation 58 gives d1, 3 4ˆr 1 1, 3 ˆr 3 1, 3 + p 1 q 1 p 3 q 3 1 α + 1,3 e 1 Λ 1+Λ 3 + κ 1 1, 3 + e 1 Λ 1 Λ 3 + κ 3 1, 3 + e 1 Λ 3 Λ 1 + κ + 1,3 e 1 Λ 1+Λ 3, 516 where α + 1,3 = T r ˆP 1 1, 3 ˆP3 1, 3 +, κ + 1,3 = κ 11, 3 + κ 3 1, 3 + β + 1,3 κ 1 1, 3 + = ˆr 11, 3, κ 3 1, 3 + = ˆr 31, 3 + p 1 q 1 p 3 q 3 and 51 Summary of the three-dromion structure The asymptotic expressions 510, 51, 513, 514, 515 and 516 all have the same form as the dromion structure given by 53 Therefore, we have shown that the three-dromion structure detu = detv [4],x decomposes asymptotically into six dromions: 4r i i,j r j i,j + p i q i p j q j 1 α i,j e 1 Λ i +Λ j +κ i i,j e 1 Λ i Λ j +κ j i,j e 1 Λ j Λ i +κ i,j di, j e 1, t, Λ i +Λ j 4ˆr i i,j ˆr j i,j + p i q i p j q j 1 α + i,j e 1 Λ i +Λ j +κ i i,j + e 1 Λ i Λ j +κ j i,j + e 1 Λ j Λ i +κ +i,je 1, t +, Λ i +Λ j for i, j {1,, 3} and i j, giving the following generalisation of Theorem :

122 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 11 Theorem 3 If detω is positive-definite, then detu, as given by 58, has the following properties: 1 detu decomposes asymptotically into six dromions as described in Theorem Each di, j decays to zero exponentially as x, y in any direction The amplitude of di, j is A := 1 α i,j p i q i p j q j β i,j + 1 as t, A + := 1 α+ i,j p i q i p j q j β + i,j + 1 as t + The amplitude is negative a as t : if α i,j > 1, b as t + : if α + i,j > 1, zero a as t : if α i,j = 1, b as t + : if α + i,j = 1, positive a as t : if α i,j < 1, b as t + : if α + i,j < 1, 3 At time t the location of di, j moves from x, y = 1 l j ξ i i, j + ξ i i, j + l i ξ j i, j + ξ j i, j + + 8l j,k t, l i,j l 1 i ξ j i, j + ξ j i, j + l 1 j ξ i i, j + ξ i i, j + + 8l i,k t, as t to x, y = 1 l j ˆξ i i, j + l ˆξ i i, j + l i ˆξ j i, j + ˆξ j i, j + + 8l j,k t, i,j l 1 i ˆξ j i, j + ˆξ j i, j + l 1 j ˆξ i i, j + ˆξ i i, j + + 8l i,k t, as t +

123 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 1 4 The trajectory of di, j changes from li,k y = x+ l j,k l 1 i l j,k + l i l i,k ξ j i, j + ξ j i, j + l 1 j l j,k + l j l i,k ξ i i, j + ξ i i, j +, l i,j l j,k as t to li,k y = x+ l j,k l 1 i l j,k + l 1 l i,k ˆξ j i, j + ˆξ j i, j + l 1 j l j,k + l j l i,k ˆξ i i, j + ˆξ i i, j +, l i,j l j,k as t Plots of dromions In this section, the interaction properties of the three-dromion structure are highlighted with various plots It is interesting to see under what conditions each dromion vanishes before and after undergoing interaction Let us label the elements of the -vectors as µi, j = ˆµi, j = a ij,1 a ij, c ij,1 c ij,, νi, j =, ˆνi, j = b ij,1 b ij, d ij,1 d ij,, µi, j + =, ˆµi, j + = a+ ij,1 a + ij, c+ ij,1 c + ij,, νi, j + =, ˆνi, j + = b+ ij,1 b + ij, d+ ij,1 d + ij, for j = 1,, 3 and i j As t, di, j vanishes T rp i i, j P j i, j + = 1 µi, j, νi, j + µi, j +, νi, j = µi, j, νi, j µi, j +, νi, j + a ij, a+ ij,1 = a ij,1 a+ ij, or b ij, b+ ij,1 = b ij,1 b+ ij, As t +, di, j vanishes T r ˆP i i, j ˆPj i, j + = 1 ˆµi, j, ˆνi, j + ˆµi, j +, ˆνi, j = ˆµi, j, ˆνi, j ˆµi, j +, ˆνi, j + c ij, c+ ij,1 = c ij,1 c+ ij, or d ij, d+ ij,1 = d ij,1 d+ ij, There is enough freedom in the parameters of the three-dromion structure to set up a situation where there are three dromions where one or two dromions vanish as t ±,,

124 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 13 Figure 54 shows a plot of the three-dromion structure with µ T 1 = 1 1, µ T = ν1 T = 3 1, ν T 7 = , µ T 3 = 1 1, 1 and ν3 T = 5 8 7, 8 so that no dromions vanish as t ± Figure 55 shows a plot of the three-dromion structure with µ T 1 = 1 3, µ T = 1 3, µ T 3 = 1 4 6, ν1 T = 1 3, ν T = 5 and ν3 T = 5 1, so that d1, and d, 3 vanish as t + Figure 56 shows a plot of the three-dromion structure with µ T 1 = 1 1, µ T = ν1 T = 3 1, ν T 7 = , µ T 3 = 1 1, 1 and ν3 T = 5 8 7, 8 so that d1, 3 vanishes as t Note that this structure has a dromion with negative amplitude Figure 57 shows the details of the interaction in Figure 56

125 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 14 a t = 01 b t = 004 c t = 005 d t = 01 Figure 54: Plot of the three-dromion structure with parameters p1 =, p =, p3 = 4, q1 = 3, q = 1, q3 = 3, r1 = and r = 1 = r3

126 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 15 a t = 015 b t = 0075 c t = 005 d t = 01 Figure 55: Plot of the three-dromion structure with parameters p1 = 1, p =, p3 = 4, q1 =, q = 1, q3 = 3 and r1 = r = r3 = 1

127 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 16 a t = 01 b t = 004 c t = 004 d t = 01 Figure 56: Plot of the three-dromion structure with parameters p1 =, p =, p3 = 4, q1 = 3, q = 1, q3 = 3, r1 = and r = 1 = r3

128 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 17 Figure 57: Details of the interaction shown in Figure 56 5 Matrix mkp single dromion As we were able to find dromions of the matrix version of the nckp equation, the matrix mkp equation should also possess dromions Again, the simplest case is the single dromion which appears from the two-soliton matrix solution obtained when n = Most of the results obtained in this section bear close resemblance to those of the matrix KP solutions Recall from Chapter 4 that the two-soliton matrix solution of ncmkp can be written in terms of f [3] = I + 1 L L, q 1 q in which and L 1 = p q 1 q 1 p 1 q q h I + p 1 A A 1, h L = p 1 q q p q 1 q 1 h 1 I + p A 1 A, h h = h 1 h q 1 q p 1 q p q 1 αp 1 p r 1 r h i = e Λ i p i r i and α = µ 1, ν µ, ν 1 p i q i q i µ 1, ν 1 µ, ν = T rp 1P Expanding detf [3] x and detf [3] and using the fact that the determinant of a projection matrix is zero and its trace is equal to its rank, we get detf [3] x = q 1 q r 1 r p 1 q p q 1 1 αh 1,x h,x h and detf [3] = 1 + h 1 h 1 p 1 q p q 1 q 1 r + h p 1 q p q 1 q r 1 +p 1 q p q 1 r 1 r + r 1 r αp 1 p q 1 q

129 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 18 We may now calculate the determinant of w, which is detw = detf [3] x detf [3] = r 1r p 1 q 1 p q 1 α q 1 q S κ S ι, 517 where S κ = e 1 Λ 1+Λ κ 1 e 1 Λ 1Λ κ e 1 Λ Λ 1 + κe 1 Λ 1+Λ, S ι = e 1 Λ 1+Λ ι 1 e 1 Λ 1Λ ι e 1 Λ Λ 1 + ιe 1 Λ 1+Λ, ι i = p i q i κ i, i = 1,, κ = κ 1 κ β and ι = ι 1 ι β The characteristics of detw, as given by equation 517, may be summarised by the following theorem: Theorem 4 If detω is positive-definite and if α 1, then detw has the following properties: 1 detw decays to zero exponentially as x, y in any direction and has a unique maximum or minimum value detw max/min = 1 αp 1 p p 1 q 1 p q β p1 p q 1 q + p 1p q 1 q + p 1 p q 1 q + p p1 q q 1 q 1 q The dromion will have negative, zero or positive amplitude The amplitude is negative if α > 1, zero if α = 1, positive if α < 1 At time t this maximum or minimum as located at x, y = 1 l 1, l ξ 1 + ξ+ 1 l 1 ξ + ξ+ + 8l,3t, where l i,j = l i i l j j l j i l i j and l j i l 1 1 ξ + ξ+ l1 ξ 1 + ξ l 1,3t, 518 = q j i pj i This result implies that the dromion is located symmetrically between the solitons in the two-soliton matrix solution 3 The trajectory of the dromion is the straight line l 1 l,3 + l l 1,3 y = l1,3 l,3 x+ ξ 1 + ξ+ 1 l 1 1 l,3 + l 1 l 1,3 ξ + ξ+ 519 l 1, l,3

130 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 19 Proof From 517, we see that detw decays to zero exponentially as x, y in any direction since, along any ray in the x, y-plane, at least one of the exponentials in the denominator is unbounded as x, y approaches infinity To see this, we use the same technique as for the matrix KP single dromion Let y = kx, where k R, be a ray in any direction Substituting this into 517 gives S κ = e 1 κ e 1 S ι = e 1 ι e 1 l 1 1 +l1 + l 1 +l k x 4 l 3 1 +l3 t l 1 l1 1 + l l 1 k x 4 l 3 l3 1 t l 1 1 +l1 + l 1 +l k x 4 l 3 1 +l3 t l 1 l1 1 + l l 1 k x 4 l 3 l3 1 t κ 1 e 1 + κe 1 ι 1 e 1 + ιe 1 l 1 1 l1 + l 1 l k x 4 l 3 1 l3 t l 1 1 +l1 + l 1 +l k x 4 l 3 1 +l3 t l 1 1 l1 + l 1 l k x 4 l 3 1 l3 t l 1 1 +l1 + l 1 +l k x 4 l 3 1 +l3 t on the denominator This expression must tend to infinity for any values of k and l j i, i, j = 1, as x ± Since detw is exponentially localized, a unique critical point must be either a maximum or minimum If we consider the conditions that α 1 and detw x and detw y vanish simultaneously, we get and in which X Y κ 1 ι 1 Y + κ ι X κ 1 κ ι 1 ι β κ + ι X Y + κ + ι κ 1 ι 1 βy = 0 X Y + κ 1 ι 1 Y κ ι X κ 1 κ ι 1 ι β κ 1 + ι 1 XY + κ 1 + ι 1 κ ι βx = 0, X = e Λ 1 and Y = e Λ Solving this equation for X and Y gives only one pair of positive roots which is e Λ 1 = κ 1 ι 1 β and e Λ = κ ι β 50 Substituting 50 into 517 gives the maximum or minimum of detw Solving 50 for x and y gives 518, the location of the dromion Eliminating t in 518 gives the trajectory of the dromion, 51 A three-dromion example For the matrix mkp solution, the determinant of the three-soliton matrix solution again gives a three-dromion structure The schematic form of the dromion scattering will be in accordance with the dromions of the matrix KP solutions as illustrated in Figure 53

131 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 130 When n = 3, expanding detf [4] x and detf [4], the expressions simplify greatly and we obtain detf [4] x = h 1,xh,x m 1,,3 + h 1,x h 3,x m 1,3, + h,x h 3,x m,3,1 h, 51 1,, 3 1 detf [4] = 1 + q 1 q q 3 h1,, 3 r 1r r 3 M + q 1 h 1 ϱ,3 + q h ϱ 1,3 + q 3 h 3 ϱ 1, r 3 h1, + p 1 q p q 1 + r h1, 3 p 1 q 3 p 3 q 1 + r 1 h, 3, 5 p q 3 p 3 q 3 where ϱ i,j = r i r j 1 α i,jq i q j p i p j, p i q j p j q i p 1 p 3 q 1 + p q 1 q + q 3 p p 3 q p 3 q 1 M = α 1,,3 p 1 q 3 p 3 q p q 1 q p 3 q 1 p p 3 q 3 + p 1 p q 1 + q 3 p 3 q 1 α 1,3, p 1 q p q 3 p 3 q 1 + α 1, p p 1 q + p 1 p q 1 p 1 q p q 1 + α,3 p3 p q 3 + p p 3 q p q 3 p 3 q, + α 1,3 p3 p 1 q 3 + p 1 p 3 q 1 p 1 q 3 p 3 q 1 m i,j,k = q i q j r i r j qk h k 1 α i,j + h k q k r k α i,j,kp k l i,k + p i l k,j l i,k l k,j α i,k,jp j l j,k + p k l k,i + α j,kp k l j,k + p j l k,j + α i,kp k l i,k + p i l k,i l j,k l k,i l j,k l k,j l i,k l k,i p k rk p iα i,j,k p jα i,k,j + α i,kα j,k q k p i p j q i q j + p jα j,k + p iα i,k, l i,j l k,j l j,k l k,i l i,k l k,i l j,k l k,j l j,k l k,j l i,k l k,i for i, j, k {1,, 3} and i j k As in the previous section, we have l i,j = p i q j To investigate the behaviour of each dromion as t ±, we fix attention on the dromion arising from the interaction of the ith and jth solitons, which will again be termed di, j Furthermore, we will call the corresponding two-soliton interaction matrix variable f i,j We consider detw = det f [4] x / detf [4] as given by 51 and 5 in a frame moving with the i, jth dromion by rewriting it in terms of q 3 j p 3 j x = ˆx + 4 q i p i q3 i p3 i q j p j q j p j qi p i q j p j q t, i p i q 3 i p 3 i y = ŷ + 4 q j p j qj 3 p3 j q i p i q j p j qi p i q j p j q t, i p i from which we obtain Λ i = q i p i ˆx + q i p i ŷ, Λ j = q j p j ˆx + q j p jŷ,

132 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 131 for i, j = 1,, 3 In accordance with the three-soliton matrix solution, we will assume, without loss of generality, that 0 > p 3 > q 3 > p > q > p 1 > q 1 Let us begin by fixing d1, With solitons 1 and fixed, h 1 and h are fixed and we study the asymptotic behaviour of h 3 as t ± We have that h 3 p 3r 3 q 3 p 3 q 3, +, t, t + When h 3 p 3r 3 q 3 p 3 q 3, equation 454 gives f 1, = b 1, A 3 + b 1,,3 A 1 A 3 + b 3,,1 A 3 A 1 + b,1,3 A A 3 + b 3,1, A 3 A 53 p3 q r 3 h + p p 3 r r 3 α,3 p3 q 1 r 3 h 1 A 1 + p 1p 3 r 1 r 3 α 1,3 A p 3 q 3 p q 3 p 3 q p 3 q 3 p 1 q 3 p 3 q 1 p p 3 r 3 α 1,3, p p 3 r 3 + A 1 A p q 3 p 3 q 1 α 1, p q 1 p 3 q 3 p 1 p 3 r 3 α 1,,3 p 1 p 3 r 3 p 3 q 3 + A A 1 + I, p 1 q 3 p 3 q α 1, p 1 q p 3 q 3 h1, p 3 q 1 q r 3 where h1, = h 1 h p r α,3 p 3 q 3 h 1 q p q 3 p 3 q p 1r 1 α 1,3 p 3 q 3 h q 1 p 1 q 3 p 3 q 1 p 1 p r 1 r α 1, + p 1p r 1 r p 3 q 3 q 1 q q 1 q p 1 q p q 1 α 1,,3 p q 1 p 1 q 3 p 3 q + α 1,3, p 1 q p q 3 p 3 q 1 To obtain the characteristics of the dromion d1,, we find the asymptotic forms of w := f 1, x f 1 1, Firstly, let us fix Λ 1 Since w is invariant under the transformation f 1, f 1, C, where C is a constant matrix, we have r 1 1, f1, = I + q 1 P 1 1, e Λ 1 + p 1 r 1 1, q 1 p 1 q 1 f + 1, = I + r 1 1,+ q 1 P 1 1, + e Λ 1 + p 1 r 1 1, + q 1 p 1 q 1 as as t, t +,

133 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 13 where P 1 1, = µ 11, ν 1 1, T µ 1 1,, ν 1 1,, P 11, + = µ 11, + ν 1 1, +T µ 1 1, +, ν 1 1, +, µ 1 1, = µ 1 p 1p 3 q 3 µ 1, ν 3 µ 3 p 3 p 1 q 3 µ 3, ν 3, ν 11, = ν 1 q 1p 3 q 3 µ 3, ν 1 ν 3 q 3 p 3 q 1 µ 3, ν 3 µ 1 1, + = µ 1 + p 1p q p 3 q 3 µ 1, ν 3 µ 3, ν p µ, ν β,3 µ 3, ν 3 p 1 q 3 p 3 q µ 1, ν p 1 q p 3 q 3 + p 1p q 3 p 3 q µ 1, ν µ, ν 3 p 3 µ 3, ν 3 β,3 µ, ν p 1 q p q 3 µ 1, ν 3 µ 3, p 1 q 3 p q ν 1 1, + = ν 1 + q 1p q p 3 q 3 µ 3, ν 1 µ, ν 3 q µ, ν β,3 µ 3, ν 3 p q 3 p 3 q 1 µ, ν 1 p q 1 p 3 q 3 + q 1p q 3 p 3 q µ, ν 1 µ 3, ν q 3 µ 3, ν 3 β,3 µ, ν p 3 q p q 1 µ 3, ν 1 ν 3, p 3 q 1 p q r 1 1, = r 1µ 1 1,, ν 1 1, µ 1, ν 1 r 1 1, + = r 1µ 1 1, +, ν 1 1, + µ 1, ν 1 = r 1 β 1,3 and = r 1 β 1,,3 β,3 These asymptotic expressions for f 1, are of the same form as the one-soliton matrix variable f [] discussed in Chapter 4 So the asymptotic forms for w are w p 1 q 1 P 4p 1 q , sech w p 1 q 1 P 4p 1 q , + sech Λ1 + ϕ 1 1, Λ1 + ϕ 1 1, + sech sech in which ϕ 1 1, = log p 1r 1 1, q 1 p 1 q 1, ϕ 1 1, + = log χ 1 1, = log r 11, p 1 q 1, χ 1 1, + = log r 11, + ξ 1 1, = log p 1 q1 1 1 r 1 1, p 1 q 1 and p 1 q 1 Λ1 + χ 1 1, Λ1 + χ 1 1, + p 1r 1 1, + q 1 p 1 q 1, as as µ ν t, t +, and the phase-constants are: ξ 1 1, + = log p 1 q1 1 1 r 1 1, + p 1 q 1 Next we fix Λ Since w is invariant under the transformation f 1, f 1, C, where C is a constant matrix, we have r 1, f1, = I + q P 1, e Λ + p r 1, q p q f + 1, = I + r 1,+ q P 1, + e Λ + p r 1, + q p q as as t t +,

134 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 133 in which P 1, = µ 1, ν 1, T µ, ν P 1, + = µ 1, + ν 1, +T µ 1, +, ν 1, +, µ 1, + = µ p p 3 q 3 µ, ν 3 µ 3 p 3 p q 3 µ 3, ν 3, ν 1, + = ν q p 3 q 3 µ 3, ν ν 3 q 3 p 3 q µ 3, ν 3 µ 1, = µ + p p 1 q 1 p 3 q 3 µ, ν 3 µ 3, ν 1 p 1 µ 1, ν 1 β 1,3 µ 3, ν 3 p q 3 p 3 q 1 µ, ν 1 p q 1 p 3 q 3 + p p 1 q 1 p 3 q 3 µ, ν 1 µ 1, ν 3 p 3 µ 3, ν 3 β 1,3 µ 1, ν 1 p q 1 p 1 q 3 µ, ν 3 µ 3, p q 3 p 1 q 1 ν 1, = ν + q p 1 q 1 p 3 q 3 µ 3, ν µ 1, ν 3 q 1 µ 1, ν 1 β 1,3 µ 3, ν 3 p 1 q 3 p 3 q µ 1, ν p 1 q p 3 q 3 + q p 1 q 1 p 3 q 3 µ 1, ν µ 3, ν 1 q 3 µ 3, ν 3 β 1,3 µ 1, ν 1 p 3 q 1 p 1 q µ 3, ν ν 3, p 3 q p 1 q 1 r 1, = r µ 1,, ν 1, µ, ν r 1, + = r µ 1, +, ν 1, + µ, ν So the asymptotic forms for w are w p q P 4p q 1 1, sech w p q P 4p q 1 1, + sech = r β 1,,3 β 1,3 = r β,3 Λ + ϕ 1, Λ + ϕ 1, + and sech sech in which ϕ 1, = log p r 1, q p q, ϕ 1, + = log χ 1, = log r 1, p q, χ 1, + = log r 1, + ξ 1, = log p q 1 1 r 1, p q and p q Λ + χ 1, Λ + χ 1, + p r 1, + q p q, as as µ 1 ν 1 t, t +, and the phase-constants are: ξ 1, + = log p q 1 1 r 1, + p q Furthermore, the soliton phase-shifts j 1, = ξ j 1, + ξ j 1,, for j = 1, are 1 1, = log β1, and 1, = log β1,, in which β1, = β 1,,3 β 1,3 β,3 The asymptotic expressions for w can now be used to describe the dromion d1, as t When h 3 p 3r 3 q 3 p 3 q 3, equation 51 and 5 give d1, r 11, r 1, + p 1 q 1 p q 1 α1, Sκ 1, Sι, 54 1,

135 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 134 where S κ 1, = e 1 Λ 1+Λ κ 1 1, e 1 Λ 1 Λ κ 1, e 1 Λ Λ 1 + κ 1, e 1 Λ 1+Λ, S ι 1, = e 1 Λ 1+Λ ι 1 1, e 1 Λ 1 Λ ι 1, e 1 Λ Λ 1 + ι 1, e 1 Λ 1+Λ, α 1, = T rp 11, P 1, +, κ 1 1, = r 11, p 1 q 1, κ 1, = r 1, + p q, ι 1 1, = p 1 q 1 κ 1 1,, ι = p q κ 1,, ι 1, = ι 11, ι 1, β 1,, and κ 1, = κ 11, κ 1, β 1, In Chapter 4, we had that L i,j = q ip j q i hi, j h j q j p i q j I + p i A j A i, for i, j {1,, 3} and i j When h 3 +, from 450 we have that L 3 0 and therefore f 1, I + L 1, q 1 + L,1, 55 q The asymptotic expression 55 is of the same form as the two-soliton matrix variable f [3] and the resulting dromion is therefore of the same form as the single dromion as given by 517 Therefore, when Λ 1 is fixed, the asymptotic forms ˆ w := f 1, x f 1 1, are ˆ w p 1 q 1 ˆP 4p 1 q , Λ1 + ˆϕ 1 1, sech ˆ w p 1 q 1 ˆP 4p 1 q , + Λ1 + ˆϕ 1 1, + sech where ˆP 1 1, = sech sech Λ1 + ˆχ 1 1, Λ1 + ˆχ 1 1, + ˆµ 11, ˆν 1 1, T ˆµ 1 1,, ˆν 1 1,, ˆP1 1, + = ˆµ 11, +ˆν 1 1, +T ˆµ 1 1, +, ˆν 1 1, +, ˆµ 1 1, = µ 1, ˆν 1 1, = ν 1, ˆµ 1 1, + = µ 1 p 1p q µ 1, ν µ p p 1 q µ, ν, ˆν 1 1, + = ν 1 q 1p q µ, ν 1 ν q p q 1 µ, ν, ˆr 11, = r 1, ˆr 1 1, + = r 1ˆµ 1 1,, ˆν 1 1, p1ˆr = r 1 β 1,, ˆϕ 1 1, 1 1, = log µ 1, ν 1 q 1 p 1 q 1 ˆχ 1 1, ˆr1 1, p1ˆr = log, ˆϕ 1 1, + 1 1, + = log and p 1 q 1 q 1 p 1 q 1 ˆχ 1 1, + ˆr1 1, + = log p 1 q 1 as as t, t +,,

136 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 135 The soliton phase-constants are ˆξ 1 1, p 1 q1 1 = log 1 ˆr 1 1, p 1 q 1 and ˆξ 1 1, + = log p 1 q1 1 1 ˆr 1 1, + p 1 q 1 When Λ is fixed, the asymptotic forms for ˆ w are ˆ w p q ˆP 4p q 1 1, Λ + ˆϕ 1, sech ˆ w p q ˆP 4p q 1 1, + Λ + ˆϕ 1, + sech in which sech sech Λ + ˆχ 1, Λ + ˆχ 1, + as as t, t +, ˆP 1, = ˆµ 1, ˆν 1, T ˆµ 1,, ˆν 1,, ˆP 1, + = ˆµ 1, +ˆν 1, +T ˆµ 1, +, ˆν 1, +, ˆµ 1, + = µ, ˆν 1, + = ν, ˆµ 1, = µ p p 1 q 1 µ, ν 1 µ 1 p 1 p q 1 µ 1, ν 1, ˆν 1, = ν q p 1 q 1 µ 1, ν ν 1 q 1 p 1 q µ 1, ν 1, ˆr 1, = r ˆµ 1,, ˆν 1, = r β 1,, µ, ν pˆr ˆr 1, + = r, ˆϕ 1, 1, = log, ˆχ 1, ˆr 1, = log, q p q p q pˆr ˆϕ 1, + 1, + = log and ˆχ 1, + ˆr 1, + = log q p q p q The soliton phase-constants are ˆξ 1, p q 1 = log 1 ˆr 1, p q and ˆξ 1, + = log p q 1 1 ˆr 1, + p q Furthermore, the soliton phase-shifts ˆ j 1, = ˆξ j 1, + ˆξ j 1,, for j = 1, are ˆ 1 1, = log β 1, + and ˆ 1, = log β 1, +, where β 1, + = β 1, where The dromion d1, as t + can now be written as d1, ˆr 11, ˆr 1, + p 1 q 1 p q 1 α 1, + S κ + 1, S ι +, 56 1, S + κ 1, = e 1 Λ 1+Λ κ 1 1, + e 1 Λ 1 Λ κ 1, + e 1 Λ Λ 1 + κ + 1, e 1 Λ 1+Λ, S + ι 1, = e 1 Λ 1+Λ ι 1 1, + e 1 Λ 1 Λ ι 1, + e 1 Λ Λ 1 + κ + 1, e 1 Λ 1+Λ, α + 1, = T r ˆP 1 1, ˆP 1, +, κ 1 1, + = κ 1, κ 1, + = κ, κ + 1, = κ+ 1 κ+ β+ 1,, ι 1 1, + = p 1 q 1 κ 1 1, + ι 1, + = p q κ 1, + and ι + 1, = ι+ 1 ι+ β+ 1,

137 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 136 Similar calculations give expressions for the dromions d, 3 and d1, 3 As t and as t + we have d, 3 r, 3 r 3, 3 + p q p 3 q 3 1 α,3 Sκ, 3Sι, 57, 3 d1, 3 r 11, 3 r 3 1, 3 + p 1 q 1 p 3 q 3 1 α1,3 Sκ, 3Sι, 58, 3 d, 3 ˆr, 3 ˆr 3, 3 + p q p 3 q 3 1 α,3 + S κ +, 3S ι +, 59, 3 d1, 3 ˆr 11, 3 ˆr 3 1, 3 + p 1 q 1 p 3 q 3 1 α 1,3 + S κ + 1, 3S ι , 3 5 Summary of the three-dromion structure The asymptotic expressions 54, 56, 57, 58, 59 and 530 all have the same form as the dromion given by 517 Therefore, we have shown that the threedromion structure detw = detf [4] x/ detf [4] decomposes asymptotically into six dromions: di, j r i i,j r j i,j + p i q i p j q j 1 α i,j Sκ i,jsι i,j ˆr i i,j ˆr j i,j + p i q i p j q j 1 α + i,j S + κ i,js + ι i,j as as t, t +, for i, j {1,, 3} and i j, giving the following generalisation of Theorem 4: Theorem 5 If detω is positive-definite, then detw, as given by 51 and 5, has the following properties: 1 detw decomposes asymptotically into six dromions as described in Theorem 4 Each di, j decays to zero exponentially as x, y in any direction The amplitude of di, j is A := A + := β i,j β + i,j 1 αi,j p ip j p i q i p j q j pi p j q i q j + p ip j q i q j + p i pj q i q j + p j pi q j q i q i qj 1 α i,j + p ip j p i q i p j q j pi p j q i q j + p ip j q i q j + p i pj q i q j + p j pi q j q i q i qj, t, t + The amplitude is negative a as t : if α i,j > 1,

138 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 137 b as t + : if α + i,j > 1, zero a as t : if α i,j = 1, b as t + : if α + i,j = 1, positive a as t : if α i,j < 1, b as t + : if α + i,j < 1, 3 At time t the location of di, j moves from x, y = 1 l j ξ i i, j + ξ i i, j + l i ξ j i, j + ξ j i, j + + 8l j,k t, l i,j l 1 i ξ j i, j + ξ j i, j + l 1 j ξ i i, j + ξ i i, j + + 8l i,k t, as t to x, y = 1 l j ˆξ i i, j + l ˆξ i i, j + l i ˆξ j i, j + ˆξ j i, j + + 8l j,k t, i,j l 1 i ˆξ j i, j + ˆξ j i, j + l 1 j ˆξ i i, j + ˆξ i i, j + + 8l i,k t, as t + 4 The trajectory of di, j changes from y = li,k x+ l j,k l 1 j l j,k + l j l i,k ξ i i, j + ξ i i, j + l 1 i l j,k + l i l i,k ξ j i, j + ξ j i, j +, l i,j l j,k as t to y = li,k x+ l j,k l 1 j l j,k + l j l i,k ˆξ i i, j + ˆξ i i, j + l 1 i l j,k + l 1 l i,k ˆξ j i, j + ˆξ j i, j +, l i,j l j,k as t +

139 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS Plots of dromions Let us again label the elements of the -vectors as µi, j = ˆµi, j = a ij,1 a ij, c ij,1 c ij,, νi, j =, ˆνi, j = for j = 1,, 3 and i j As t, di, j vanishes b ij,1 b ij, d ij,1 d ij,, µi, j + =, ˆµi, j + = a+ ij,1 a + ij, c+ ij,1 c + ij,, νi, j + =, ˆνi, j + = b+ ij,1 b + ij, d+ ij,1 d + ij,,, T rp i i, j P j i, j + = 1 µi, j, νi, j + µi, j +, νi, j = µi, j, νi, j µi, j +, νi, j + a ij, a+ ij,1 = a ij,1 a+ ij, or b ij, b+ ij,1 = b ij,1 b+ ij, As t +, di, j vanishes T r ˆP i i, j ˆPj i, j + = 1 ˆµi, j, ˆνi, j + ˆµi, j +, ˆνi, j = ˆµi, j, ˆνi, j ˆµi, j +, ˆνi, j + c ij, c+ ij,1 = c ij,1 c+ ij, or d ij, d+ ij,1 = d ij,1 d+ ij, Figure 58 shows a plot of the three-dromion structure with µ T 1 = 1, µ T = 3 6, µ T 3 = 1 ν1 T =, ν T = 1 3 and ν3 T = ,, so that d1, 3 vanishes as t and d1,, d, 3 vanish as t +

140 CHAPTER 5 DROMIONS OF THE MATRIX EQUATIONS 139 a t = 04 b t = 05 c t = 05 d t = 04 Figure 58: Plot of the three-dromion structure with parameters p1 = 9, p = 5, p 3 = 1 3, q 1 = 6, q = 3, q3 =, r1 = r = r3 = 1