On Kaup-Kupershchmidt type equations and their soliton solutions

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1 IL NUOVO CIMENTO 8 C 2015) 161 DOI 10.19/ncc/i Colloquia: VILASIFEST On Kaup-Kupershchmidt type equations and their soliton solutions V. S. Gerdjikov Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences 72 Tzarigradsko chaussee, Blud., 1784 Sofia, Bulgaria received 11 January 2016 Summary. We start with the Lax representation for the Kaup-Kupersschmidt equation KKE). We outline the deep relation between the scalar Lax operator and the matrix Lax operators related to Kac-Moody algebras. Then we derive the MKdV equations gauge equivalent to the KKE. Next we outline the symmetry and the spectral properties of the relevant Lax operator. Using the dressing Zakharov- Shabat method we demonstrate that the MKdV and KKE have two types of onesoliton solutions and briefly comment on their properties. PACS 0.75.Lm Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations. PACS Tg Optical solitons; nonlinear guided waves. PACS 52.5.Sb Solitons; BGK modes. PACS Ap Basis sets LCAO, plane-wave, APW, etc.) and related methodology scattering methods, ASA, linearized methods, etc.). 1. Introduction In 1980 D. Kaup and Satsuma [19,26] analyzed integrable NLEE related to the scalar third-order operator: 1) Lψ ψ +6Qx, t) ψ x x +6Rx, t) λ )ψ =0, and demonstrated that it allows two interesting reductions each giving rise to an interesting generaliztion of KdV [7]: 2) A) R =0, B) R = 1 Q 2 x. Creative Commons Attribution 4.0 License 1

2 2 V. S. GERDJIKOV In the first case the operator L A combined with ) M A ψ ψ ) t 9λ 18 2 Q 2 ψ 2 ) x 2 x 2 6 Q ψ x 2 6Q2 x 6λ Qψ =0, leads to the Sawada-Kotera equation [26, 19, 27] 4) Q t + 5 Q ) x 5 +0 Q x Q + 2 Q Q x Q x x Q2 =0. Here and below we have replaced Kaup s spectral parameter λ by λ. In the second case the Lax operator L B with 5) M B ψ ψ t 9λ 2 ψ x 2 + provide the Lax representation for 6) 2 ) Q ψ x 2 +12Q2 x Q x +12λ Qλ +24 Q ) x Q ψ =0, Q t + 5 Q x 5 +0 Q x Q ) Q Q 2 x Q x x Q2 =0, known today as the Kaup-Kuperschmidt equation [19, 20]. It is easy to check that there is no elementary change {ψ, x, t} { ψ, x, t} which could transform eq. 4) into 6). Both these equations are inequivalent also to the KdV5 equation, which takes the form 7) Q t + 5 Q ) x 5 +0 Q x Q Q Q +2 2 x Q x x Q2 =0. The answer to the question why these three equations, so similar, are inequivalent was soon discovered. It can be traced back to two seminal papers. The first one is by Mikhailov [21] who introduced the notion of the reduction group and discovered the integrability of the 2-dimensional Toda field theories TFT). In fact, the eqs. 4) and 6) belong to the hierarchies of NLEE containing 2-dimensional TFT related to the algebra sl) but requiring different inequivalent reductions. In the other important paper Drinfeld and Sokolov [9] demonstrated the deep relations between the scalar operators of the form 1) and the first-order matrix ordinary differential operators with deep reductions: 8) Lψ i ψ + Ux, t, λ)ψx, t, λ) =0, x Ux, t, λ) =qx, t) λj), related to the Kac-Moody algebras. In fact, imposing the condition that Ux, t, λ) = qx, t) λj belongs to a certain Kac-Moody algebra is equivalent to imposing on Ux, t, λ) the relevant reduction in the sense of Mikhailov. However, even until now the KKE and Sawada-Kotera equations still attract attention, especially concerning the construction and the properties of their solutions, see [1, 5, 8, 10, 11, 22-25] The paper is organized as follows. In sect. 2 we describe the gauge transformations that underly the deep relation between the scalar Lax operator 1) and its equivalent 14)

3 ON KAUP-KUPERSHCHMIDT TYPE EQUATIONS AND THEIR SOLITON SOLUTIONS and 22). In sect. we construct the Lax representation for the MKdV equations gauge equivalent to KKE. In the next sect. 4 we introduce the fundamental analytic solutions FAS) of L 14) and the relevant Riemann-Hilbert problem which underlies the inverse scattering problem for L. In sect. 5 we apply the dressing Zakharov-Shabat method [5] for deriving the soliton solutions of the MKdV. In sect. 6 we demonstrate that the poles of the dressing factor and its inverse are in fact discrete eigenvalues of the Lax operator L. We end with discussion and conclusions. 2. Preliminaries From scalar to matrix operators. First we will demonstrate how one can relate to a scalar ordinary differential operator a first-order matrix ordinary operator [9]. We will work this out on the example of the operator L 1). Indeed, let us consider the - component vector ψx, t, λ) =ψ xx +6Qψ, ψ x,ψ) T and let us differentiate it with respect to x. Using eq. 1) one easily finds that the third order scalar operator L is equivalent to 9) L 1) ψx, t, λ) = ψ x + Ũ1)x, t, λ) ψx, t, λ) =0, q 1) x, t) = 0 0 6R Q x ) 0 0 6Q Ũ 1) x, t, λ) = q 1) x, t) J 1) λ),, J 1) λ) = 0 0 λ Similarly any ordinary scalar differential operator of order n can be rewritten as a first order n n operator of special type: all x-dependent coefficients occupy just the last column of q 1) x, t), see [9] The relation between the matrix operators and Kac-Moody algebras. The next important step was proposed by Drinfeld and Sokolov [9]. They proposed to apply to L a gauge transformation [4] so that the new operator L acquires canonical form from the point of view of Kac-Moody algebras. This we will do in two steps. The first step takes L 1) to the operator L 2) with potential 10) Ũ 2) x, t, λ) =g 1 Ũ 1) x, t, λ)g + g 1 g x, where gx, t) is a upper-triangular matrix of the form 11) gx, t) = 1 q 1 c 0 1 q 2. The constraints that we impose on q 1, q 2 and c require that 12) Ũ 2) x, t, λ) =q 2) x, t) J 1) λ),

4 4 V. S. GERDJIKOV where q 2) x, t) is a diagonal matrix. This can be achieved imposing: 1) cx) =q 1 q 2 q 2 1 q 1 x, 6Qx) = q 1 x q 2 x q2 1 q q 1 q 2., 6Rx) =q 1 q 2 q 1 q 2 )+ q 1 x q q 2 2 2q 2 x 2 q 2 x 2 With this choice for Q and R we find q 2) x, t) = diag q 1, q 2 q 1, q 2 ). Note that tr qx, t)) = 0, i.e. it belongs to the Cartan subalgebra of the Lie algebra sl). The second step is to apply to L 2) a similarity transformation by the diagonal matrix C 2 λ) = diag λ 1, 1,λ). Thus we obtain the operator 14) Ũx, L = C 2 λ) L 2) C2 1 + Ũx, t, λ), x q qx, t) = 0 q 2 q 1 0, 0 0 q 2 t, λ) = qx, t) λ J, J = Similar transformations can be applied also to the M-operator in the Lax pair. 2.. Relation to Kac-Moody algebras. Let us now explain the relation of the above operator L 14) to the Kac-Moody algebras [18]. Skipping the details we will outline the construction of the Kac-Moody algebras. Here we will assume that the reader is familiar with the theory of simple Lie algebras [17]. In fact it will be enough to know the Cartan-Weyl basis of the algebra sl), in Cartan classification this algebra is denoted as A 2. The algebra sl) has rank 2 and its root system contains three positive roots: Δ + = {e 1 e 2,e 2 e,e 1 e } and three negative roots Δ = { e 1 + e 2, e 2 + e, e 1 + e }. The first two positive roots α 1 = e 1 e 2 and α 2 = e 2 e form the set of simple roots of sl). The third positive root is α max = e 1 e is the maximal one. We will say that the simple roots are of height 1, the maximal root α max = α 1 +α 2 is of height 2. Similarly, the negative roots α 1 and α 2 have height 1 and the minimal root α min = α max has height 2 = 1 mod ). The Cartan-Weyl basis of sl) is formed by the Cartan subalgebra h and by the Weyl generators E α and E α, α Δ +. In the typical representation these are given by 15) H α1 = E α2 = , H α2 =, E αmax = , E α1 =, E α = Eα T , The main tool in constructing the Kac-Moody algebra based on sl) is the grading, which according to [18] must be performed with the Coxeter automorphism. In our case the grading consists in splitting the algebra sl) into the direct sum of three linear

5 ON KAUP-KUPERSHCHMIDT TYPE EQUATIONS AND THEIR SOLITON SOLUTIONS 5 subspaces as follows: 16) sl) g 0) g 1) g 2), g 0) span {H α1,h α2 }, g 1) span {E α1,e α2,e αmin }, g 2) span {E α1,e α2,e αmax }. In other words if we assume that the Cartan generators have height 0, then each of the subspaces g k) consists of elements of height k modulo, which is the Coxeter number of sl). The important property of the subspaces g k) is provided by the relation: if X k g k mod ) and X m g m mod ) then 17) [X k,x m ] g k+m mod ). Next the elements of the Kac-Moody algebra based on this grading of sl) consists of finite or semi-infinite series of the form 18) Xλ) = X p λ p, X p g p mod ). p N The subspaces g p) are in fact the eigensubspaces of the Coxeter automorphism C 0 which in this case is an element of the Cartan subgroup of the form ) 2πi 19) C0 =exp H ρ, ρ = e 1 e ; i.e. C0 = diag ω, 1,ω 1 ). Indeed, it is easy to check that C 0 E α C 1 0 = ω k E α, where ω = e 2πi/ and k =htα). Obviously C 0 = 1. Remark 1. In fact we will use also an alternative grading, used also in [9] in which the subspaces g 1) and g 2) are interchanged. It is generated by an equivalent realization of the Coxeter automorphism: C 1 0 E C α 0 = ω k E α,wherek = ht α). Remark 2. The gauge transformation described above is the analogue of the famous Miura transformation, which maps the KdV equation into the modified KdV MKdV) equation. Therefore the Lax pair 2) will produce not the KKE, but rather a system of MKdV eqs. that are gauge equivalent to KKE. In order to understand the interrelation between the Kac-Moody algebras and the ordinary differential operators it remains to note that the potential Ũx, t, λ) in 14) is an element of the Kac-Moody algebra A 1) 2 with the the alternative grading, see Remark Factorized ordinary differential operators. In fact the matrix operator L 14) can be brought back to scalar form. Indeed, the matrix scattering problem Lχx, t, λ) =0 can be written down as 20) χ 1 x + q 1x, t)χ 1 x, t, λ) =λχ x, t, λ), χ 2 x +q 2x, t) q 1 x, t))χ 2 x, t, λ) =λχ 1 x, t, λ), χ x q 2x, t)χ x, t, λ) =λχ 2 x, t, λ),

6 6 V. S. GERDJIKOV which can easily be rewritten as the following scalar eigenvalue problem: 21) Lχ = λ χ x, t, λ), L =D x + q 1 x, t))d x +q 1 x, t) q 2 x, t))d x q 2 x, t)), which is a factorized third-order differential operator L. Thus the gauge transformation 10) effectively takes the scalar operator L 1) into the factorized one L 21).. Lax representation of MKdV equations. 1. The generic two-component MKdV s equations. Now we have the tools to construct the Lax pair of the MKdV equation which is gauge equivalent to KKE. In this section we will derive the Lax representation of the fifth order MKdV equations. This derivation, as well as the direct and inverse scattering problems for these Lax operators are more conveniently executed if the term λ J is taken in diagonal form. This is easily achieved by a similarity transformation with the constant matrix w 0 : 22) L = w0 1 Lw 0, w 0 = 1 ω 1 ω ω 2 1 ω Thus we construct the Lax pair for the MKdV eqs. as follows: 2) Lψx, t, λ) i ψ +qx, t) λj) ψx, t, λ) =0, x Mψx, t, λ) i ψ t + V x, t, λ) λ 5 K ) ψx, t, λ) =0, 4 V x, t, λ) = V s x, t)λ s, J = diag ω, 1,ω 2 ), K = diag ω 2, 1,ω), s=0 where the basis in the algebra sl) is given in the Appendix. Below we use the notations: 24) qx, t) =i q 1 B 0) 1 + q 2 B 0) 2 ), J = ω2 B 1), K = aωb2) q 1 = ωq 1 + ω 1 q 2, q 2 = ω 1 q 1 + ωq 2 ),, and ω =exp2πi/).. 2. Solving the recurrent relations and Λ-operators. The condition [L, M] = 0 must hold true identically with respect to λ. This leads to a set of equations: 25) λ 5 [K, q] =[J, V 4 ], λ k i V k x +[q, V k]=[j, V k 1 ], k =1,...,4; λ 0 i V 0 x i q t =0. Equations 25) can be viewed as recurrent relations which allow one to determine V k in terms of q and its x-derivatives. This kind of problems have been thoroughly analyzed,

7 ON KAUP-KUPERSHCHMIDT TYPE EQUATIONS AND THEIR SOLITON SOLUTIONS 7 see [14, 1], so below we just briefly mention the effects of the Z reductions we have imposed. Obviously we need to split each od the coefficients V k g k) into diagonal and offdiagonal parts: 26) V k = Vk f + w k B k), k = k mod. From the appendix it is clear that only B 1) and B 2) are non-vanishing, while, for example, V 0 and V do not have diagonal parts, so V 0 V0 f and V V. f The linear mapping ad J [J, ] is also playing an important role. The diagonal and the off-diagonal parts of the matrices in fact provide the kernel and the image of ad J. So in the space of off-diagonal matrices we can define also the inverse ad 1 J. In addition ad J and ad 1 J map the linear spaces g k) as follows: 27) ad J : g k) g k+1), ad 1 J : gk) g k 1). The formal solution of the recurrent relations 25) is most conveniently written down with the help of the recursion operators: 28) Λ 0 Y =ad 1 J i Y x Λ k X f k =ad 1 J +[q, Y ]f ), Y g 0), i Xf k x +[q, Xf k] f + i x [q, Bk) ] dy ) [qy),xk],b f k), where X k g k). Thus the formal solution of the recurrent relations provides the following answer for V k : 29) V 4 =ad 1 J [K, q], f V =Λ 0 V 4, V2 f =Λ 2 V f + i w 2B 2), V 1 f =Λ 1 V2 f + i w 1B 1), V 0 f =Λ 0 V1, f where 0) w k = x dy [qy),v f k],b k). At the end we get also the formal expression for the corresponding NLEE: 1) i q t + a x ΛΛ 0ad 1 J [K, q] =0, Λ =Λ 0Λ 1 Λ 2. Obviously, one can consider as a potential to the M-operator polynomial V N) = N p=0 V pλ p of any power N as long as N +1 0 mod. The corresponding NLEE will be generated by a relevant polynomial of the recursion operators Λ k.

8 8 V. S. GERDJIKOV.. The explicit form of the M-operator. Let us now do the calculations for the V k explicitly. Skipping the details we have 2) ) 4) 5) V 4 x, t) = ia q 1 ω 2 B 1) 1 + q 2 B 1) 2 ), V x, t) =av,1 B 0) 1 v,2 B 0) 2, v,1 = q 1 x +q2 2, v,2 = q 2 x q2 1, V 2 x, t) = ia 9 v 2,1ωB 2) 1 + v 2,2 B 2) 2 + v 2, B 2) ), v 2,1 = 2 q 1 x 2 +6q q 2 2 x, v 2,2 = 2 q 2 x 2 6q q 1 1 x, q 2 v 2, =q 1 x q q 1 2 x +6q 1 + q2). V 1 x, t) = a v 1,1ω 2 B 1) 1 + v 1,2 B 1) 2 + v 1, ω 2 B 1) ), v 1,1 = q 1 x +q 2 q 2 2 x 2 +6 q2 x ) 2 q 1 +27q 1 q 2 x 9q2 1 q 2 x 18q 1q 1 + q 2), 6) v 1,2 = q 2 x q 1 v 1, = 2 q 1 x 2 6 q1 x ) 2 q 2 +27q 1 q 2 x q 1 9q2 2 2 q 1 q 2 x 2 + q 2 q 2 1 x 2 q 1 q 2 x x q 1 q 2 q2 1 x +q2 2 x +9q2 1q2 2 V 0 x, t) = ia 9 v 0,1ωB 0) 1 + v 0,2 B 0) 2 ), v 0,1 = 4 q 1 x q 2 2 q 2 x x 2 +45q 2 q1 x ) 2 + q 1 2 q 1 x 2 ) x 18q 2q1 + q2), ). + 45q1 + q2) q 1 x +27q2 25q1 +2q2), q2 v 0,2 = 4 q 2 x 4 15 q 1 2 ) ) 2 q 1 x x 2 +45q 2 q q 2 x x 2 The NLEE: 45q 1 + q 2) q 2 x +27q2 12q 1 +5q 2), q 1 7) t + a v 01 9 x =0, q 2 t + a v 02 9 x = Special reductions. Kaup considers two special reductions on his La operator L: A) R =0andB)R = 1 2 Q x. It is the second reduction that is responsible for the KKE; in terms q 1 and q 2 it can be formulated as 8) q 2 = q 1. It may be realized using external automorphism of sl), so it must be responsible for A 2) 2 Kac-Moody algebra.

9 ON KAUP-KUPERSHCHMIDT TYPE EQUATIONS AND THEIR SOLITON SOLUTIONS 9 9) Imposing the reduction 8) we get the equation q 1 t = a 4 q 1 9 x x q 1 2 q 1 x x 2 45q 1 which is gauge equivalent to the KKE. q1 4. The FAS of the Lax operators with Z -reduction. x ) 2 + q 1 2 q 1 x 2 ) ) +81q1 5, The idea for the FAS for the generalized Zakharov-Shabat GZS) system has been proposed by Shabat [28], see also [29]. However for the GZS J is with real eigenvalues, while our Lax operator has complex eigenvalues. The ideas of Shabat were generalized by Beals and Coifman [6] and Caudrey [2] for operators L related to the algebras sln); these results were extended to L operators related to any simple Lie algebra i [14], see also [12, 1, 15, 16]. The Jost solutions of eq. 2) are defined by 40) lim x φ +x, λ)e iλjx = 1, lim φ x, λ)e iλjx = 1, x They satisfy the integral equations: x 41) Y ± x, λ) =1 + dye iλjx y) Qy)Y ± y, λ)e iλjx y), ± where Y ± x, λ) =φ ± x, λ)e iλjx. Unfortunately, with our choice for J = diag ω, 1,ω 2 ) this integral equations have no solutions. The reason is that the factors e iλjx y) in the kernel in 41) can not be made to decrease simultaneously. Following the ideas of Caudrey, Beals and Coifman, see [2,6,14] we start with the Jost solutions for potentials on compact support, i.e. assume that qx) =0forx< L 0 and x>l 0. Then the integrals in 41) converge and one can prove the existence of Y ± x, λ). The continuous spectrum of L consists of those points λ, for which e iλj k J j)x y) oscillate, which means that 42) Im λj k J j )=Imλω 2 k ω 2 j )=0. It is easy to check that for each pair of indices k j eq. 42) has a solution of the form arg λ = const depending on k and j. The solutions for all choices of the pairs k, j fill up a pair of rays l ν and l ν+ which are given by 4) l ν :argλ) = π2ν +1), Ω ν : 6 π2ν +1) 6 arg λ π2ν +), 6 where ν =0,...,5, see fig. 1. Thus the analyticity regions of the FAS are the 6 sectors Ω ν, ν =0,...,5 split up by the set of rays l ν, ν =0,...,5, see fig. 1. Now we will outline how one can construct a FAS in each of these sectors. Obviously, if Im λαj) =0ontheraysl ν l ν+, then Im λαj) > 0forλ Ω ν Ω ν+1 Ω ν+2 and Im λαj) < 0forλ Ω ν 1 Ω ν 2 Ω ν ; of course all indices here

10 10 V. S. GERDJIKOV l 1 λ b 1 b 0 l l 0 + b 2 b 5 + l + + l 5 b b 4 l 4 Fig. 1. The contour of the RHP with Z -symmetry fills up the rays l ν, ν =1,...,6. By and respectively, by + and ) we have denoted the locations of the discrete eigenvalues corresponding to a soliton of first type respectively, of second type). are understood modulo 6. As a result the factors e iλjx y) will decay exponentially if Im αj) < 0andx y>0orifimαj) > 0andx y<0. In eq. 44) below we have listed the signs of Im αj) for each of the sectors Ω ν. To each ray one can relate the root satisfying Im λαj) =0,i.e. 44) l 0, ±e 1 e 2 ) Ω 0 α 1 < 0, α 2 > 0 α > 0, l 1, ±e 1 e ) Ω 1 α 1 > 0, α 2 > 0 α < 0, l 2, ±e 2 e ) Ω 2 α 1 < 0, α 2 < 0 α < 0. There are two fundamental regions: Ω 0 and Ω 1. The FAS in the other sectors can be obtained from the FAS in Ω 0 and Ω 1 by acting with the automorphism C 0 : 45) C 0 Ω ν Ω ν+2, C 0 l ν l ν+2, ν =0, 1,...,5. The next step is to construct the set of integral equations for FAS which will be analytic in Ω ν. They are different from the integral equations for the Jost solutions 41) because for each choice of the matrix element k, j) we specify the lower limit of the integral so that all exponential factors e iλj k J j)x y) decrease for x, y ±, x 46) Xkjx, ν λ) =δ kj + i ɛ kj dye iλj k J j)x y) h q kp y)xpjy, ν λ), where the signs ɛ kj for each of the sectors Ω ν are collected in the table I, see also [0,12,16]. We also assume that for k = jɛ kk = 1. p=1

11 ON KAUP-KUPERSHCHMIDT TYPE EQUATIONS AND THEIR SOLITON SOLUTIONS 11 Table I. The set of signs ɛ kj for each of the sectors Ω ν. k, j) 1,2) 1,) 2,) 2,1),2),1) Ω Ω Ω Ω Ω Ω The solution of the integral equations 46) will be the FAS of L in the sector Ω ν. The asymptotics of X ν x, λ) andx ν 1 x, λ) along the ray l ν can be written in the form [14, 16]: 47) lim x eiλjx X ν x, λe i0 )e iλjx = S ν + λ), λ l ν, lim x eiλjx X ν x, λe i0 )e iλjx = Tν λ)d ν + λ), λ l ν, lim x eiλjx X ν 1 x, λe i0 )e iλjx = Sν λ), λ l ν, lim x eiλjx X ν 1 x, λe i0 )e iλjx = T ν + λ)dν λ), λ l ν, where the matrices S ± ν and T ± ν belong to su2) subgroups of sl). More specifically from the integral equations 46) we find 48) S 0 + λ) =1 + s+ 0;21 E 21, T0 λ) =1 + τ 0;12 E 12, S0 λ) =1 + s+ 0;12 E 12, T 0 + λ) =1 + τ 0;21 + E 21, D 0 + λ) =d+ 0;1 E d + E 22 + E, D0 λ) = 1 0;1 d E 11 + d 0;1 E 22 + E, 0;1 and 49) S 1 + λ) =1 + s+ 1;1 E 1, T1 λ) =1 + τ 1;1 E 1, S1 λ) =1 + s+ 1;1 E 1, T 1 + λ) =1 + τ 1;1 + E 1, D 1 + λ) =d+ 1;1 E 11 + E d + E, D1 λ) = 1 1;1 d E 11 + E 22 + d 1;1 E. 1;1 By E kj we mean a matrix with matrix elements E kj ) mn = δ um δ jn. The corresponding factors for the asymptotics of X ν x, λe i0 )forν>1are obtained from eqs. 48), 49) by applying the automorphism C 0. If we consider potential on finite support, then we can define not only the Jost solutions Ψ ± x, λ) but also the scattering matrix T λ) = φ x, λ)φ 1 + x, λ). The factors S ν ± λ), T ν ± λ) andd ν ± λ) provide an analog of the Gauss decomposition of the scattering matrix with respect to the ν-ordering, i.e. 50) T ν λ) =T ν λ)d + ν λ)ŝ+ ν λ) =T + ν λ)d ν λ)ŝ ν λ), λ l ν.

12 12 V. S. GERDJIKOV The Z n -symmetry imposes the following constraints on the FAS and on the scattering matrix and its factors: 51) C 0 X ν x, λω)c0 1 = X ν 2 x, λ), C 0 T ν λω)c0 1 = T ν 2 λ), C 0 S ν ± λω)c0 1 = S ν 2 ± λ), C 0D ν ± λω)c0 1 = D ν 2 ± λ), where the index ν 2 should be taken modulo 6. Consequently we can view as independent only the data on two of the rays, e.g. on l 0 and l 1 ; all the rest will be recovered using the reduction conditions. If in addition we impose the Z 2 -symmetry, then we will have also 52) a) K0 1 x, λ )) K 0 = ˆX N+1 ν x, λ), K0 1 ν λ ))K 0 = Ŝ N+1 ν λ), b) K0 1 x, λ )) K 0 = ˆX ν x, λ), K0 1 ν λ ))K 0 = Ŝ N+1 ν λ), where K 0 = E 1, +E 2,2 +E,1 and by the hat we denote the inverse matrix. Analogous relations hold true for T ± ν λ) andd ± ν λ). One can prove also that D + ν λ) respectively, D ν λ)) allows analytic extension for λ Ω ν respectively, for λ Ω ν 1. Another important fact is that D + ν λ) =D ν+1 λ) for all λ Ω ν. The next important step is the possibility to reduce the solution of the ISP for the GZSs to a local) RHP. More precisely, we have 5) X ν x, t, λ) =X ν 1 x, t, λ)g ν x, t, λ), λ l ν, G ν x, t, λ) =e iλjx+λ5kt) G 0,ν λ)e iλjx+λ5kt), G 0,ν λ) =Ŝ ν S ν + λ). t=0 The collection of all these relations for ν =0, 1,...,5 together with 54) lim λ Xν x, t, λ) =1, can be viewed as a local RHP posed on the collection of rays Σ {l ν } 2N ν=1 with canonical normalization. Rather straightforwardly we can prove that if X ν x, λ) is a solution of the RHP then χ ν x, λ) =X ν x, λ)e iλjx is a FAS of L with potential 55) qx, t) = lim λ J X ν x, t, λ)j ˆX ) ν x, t, λ). λ 5. The dressing method and the N-soliton solutions The main idea of the dressing method [5, 21, 21] is, starting from a known regular solution of the RHP X ν 0 x, t, λ) to construct a new singular solution X ν 1 x, t, λ) ofthe same RHP. The two solutions are related by a dressing factor ux, t, λ) 56) X ν 1 x, t, λ) =ux, t, λ)x ν 0 x, t, λ), which may have pole singularities in λ. A typical anzats for ux, t, λ) is given by [21], see also [, 4, 14, 1]: 57) ux, t, λ) =1 + 2 s=0 N1 l=1 C s A l C s N λ λ l ω s + r=n 1+1 C s A r C s N λ λ r ω s + r=n 1+1 ) C s A rc s λ λ r)ω s,

13 ON KAUP-KUPERSHCHMIDT TYPE EQUATIONS AND THEIR SOLITON SOLUTIONS 1 with N 1 +6N 2 poles and λ p is real if p 1,N 1 and complex if p N 1 +1,N 1 + N 2. In [] both types of simplest one-soliton solutions for the Tzitzeica equation are derived. Note that Tzitzeica equation possesses Lax representation with the same Lax operator, but its M-operator is linear with respect to λ 1. The dressing factor ux, t, λ) satisfies the equation: 58) i u x +q1) x, t) λj)ux, t, λ) ux, t, λ)q 0) x, t) λj) =0, where q 0) x, t) typically chosen to be vanishing) corresponds to the naked Lax operator or to the regular solution of RHP), while q 1) x, t) is the potential of the dressed Lax operator. The eq. 58) must hold true identically with respect to λ. We also assume that the residues A k x, t) are degenerate matrices of the form: 59) A k x, t) = n k x, t) m T k x, t), A k ) ij x, t) =n k;i x, t)m k;j x, t). Thus ux, t, λ) forn 1 = N 2 = 1 has 9 poles located at λ 1 ω k with λ 1 real and λ 2 ω k, λ 2ω k, with k =0, 1, 2andλ 2 complex. Evaluating the residue of eq. 58) for λ = λ k one finds that the polarization vectors n k x, t) and m T k x, t) must satisfy the equations: 60) i n k x + q1) x, t) λ k J) nk x, t) =0, i mt k x + λ k m T k x, t) J =0, where we have put q 0) x, t) = 0. Then the vectors m T k x, t) must depend on x and t as follows: 61) m 1 = μ 2 e 2X1 +2 μ 1 e X1 cos Ω 1 2π ), m 2 = μ 2 e 2X1 +2 μ 1 e X1 cos Ω 1 ) m = μ 2 e 2X1 +2 μ 1 e X1 cos Ω 1 + 2π ), X 1 = 1 2 λ 1x + λ 5 1t), Ω 1 = 2 λ 1x λ 5 1t) α 1, μ 1 = μ 1 e iα1. The factors ux, t, λ) must also satisfy all symmetry conditions characteristic for the FAS. The Z symmetry is already taken into account with the anzatz 57). From the second Z 2 -reduction 52), K0 1 u x, t, λ )K 0 = u 1 x, t, λ), after taking the limit λ λ k, we obtain algebraic equation for n k in terms of m T k : Below we list the relevant formulae just for the two types of one-soliton solutions for the general case see [21, ]): 62) a) n 1 = A 1 m 1, b) n2 n 2 ) D F = F D ) 1 m2 m 2 ), where case a) corresponds to the choice N 1 =1,N 2 = 0 while case b) is relevant for

14 14 V. S. GERDJIKOV N 1 =0,N 2 = 1. The notation above are as follows: 6) A = 1 F = 2λ 1 diag Q 1),Q 2),Q ) ), D = 1 1 λ 2 + λ, 2 diag K 1),K 2),K ) ), 2λ 2 diag P 1),P 2),P ) ), Q j) = m T 1 Λ j) 11 λ l,λ 1 ) m 1, K j) = m,t 2 Λ j) 12 λ 1,λ 2) m 1, P j) = m T 2 Λ j) 21 λ 2,λ 1 ) m l, with 64) Λ j) lp = λ lλ p E 1+j, j + λ 2 l E 2+j,2 j + λ 2 pe +j,1 j, j =1, 2,. Skipping the details, we just mention that this approach allows one to obtain the explicit form of the N-soliton solutions. We just mention that along with the explicit expressions for the vectors n k in terms of m j that follow from eqs. 62) 64) and take into account that m j are solutions of the naked Lax operator with vanishing potential q 0) =0. We end this section with a few comments about the simplest one-soliton solutions of the MKdV and KKE equations. The first one is these one-soliton solutions with generic choice of the polarization vectors are not traveling waves. Skipping the details see e.g. []) we find for the naked solution of the Lax operator ) ) n m 2m1 m 65) q 1 x) = x ln 1 = x ln λ 1 m Note that the soliton solution of the KKE can be obtained from 65) by 66) 6Q = 2q 1,x q 2 1. In the special case μ 2 =0wehave 1 67) q 1 x, t) = x ln 2 + ) 2 tan2 Ω 1 = λ 1 tan Ω 1 2sin 2 Ω This solution is obviously singular. operator In addition the relevant potential of the Lax 68) 6Q = 9λ2 11 4sin 2 Ω 1 ) sin 2 Ω 1 ) 2 is not in the functional class, since it does not decay to zero. A possible way to find regular soliton solutions of these equations is to take into account the fact that both MKdV and KKE are invariant under the transformation x ix, t it, Q Q.

15 ON KAUP-KUPERSHCHMIDT TYPE EQUATIONS AND THEIR SOLITON SOLUTIONS 15 Table II. The set of signs t k) ν for each of the sectors Υ ν 7). Υ 0 Υ 1 Υ 2 Υ Υ 4 Υ 5 t 1) ν t 2) ν t ) ν Then Ω 1 iω 1 andweget 69) q 1 x, t) = λ 1 tanh Ω 1 2sinh 2 Ω 1 1, 6Q = 9λ sinh 2 Ω 1) 1 2sinh 2 Ω 1 )2, which this time is singular only at two points sinh Ω 1 = ± 1 2. Moreover the potential Q x, t) decays for x ± 6. The resolvent of the Lax operator The FAS can be used to construct the kernel of the resolvent of the Lax operator L. In this section by χ ν x, λ) we will denote 70) χ ν x, λ) =ux, λ)χ ν 0x, λ), where χ ν 0x, λ) is a regular FAS and ux, λ) is a dressing factor of general form 57). Remark. The dressing factor ux, λ) has N 1 +6N 2 simple poles located at λ l ω p, λ r ω p and λ rω p where l =1,...,N 1, r =1,...,N 2 and p =0, 1, 2. Its inverse u 1 x, λ) has also N 1 +6N 2 poles located λ l ω p, λ r ω p and λ rω p. In what follows for brevity we will denote them by λ j, λ j for j =1,...,N 1 +6N 2. Let us introduce 71) 72) R ν x, x,λ)= 1 i χν x, λ)θ ν x x )ˆχ ν x,λ), ) Θ ν x x ) = diag t 1) ν θt 1) ν x x )),t 2) ν θt 2) ν x x )),t ) ν θt ) ν x x )), where θx x ) is the step-function and t k) ν = ±1, see table II. Theorem 1. Let Qx) be a Schwartz-type function and let λ ± j dressing factor ux, λ) 57). Then be the simple zeroes of the 1. The functions R ν x, x,λ) are analytic for λ Υ ν where 7) b ν :argλ = πν +1), Υ ν : having pole singularities at ±λ ± j ; πν +1) arg λ 2. R ν x, x,λ) is a kernel of a bounded integral operator for λ Υ ν ; πν +2).

16 16 V. S. GERDJIKOV. R ν x, x,λ) is uniformly bounded function for λ b ν and provides a kernel of an unbounded integral operator; 4. R ν x, x,λ) satisfy the equation: 74) Lλ)R ν x, x,λ)=1δx x ). Proof 1 Idea of the proof). 1. First we shall prove that R ν x, x,λ) has no jumps on the rays l ν. From sect. we know that X ν x, λ) and therefore also χ ν x, λ) are analytic for λ Ω ν. So we have to show that the limits of R ν x, x,λ) for λ l ν from Υ ν and Υ ν 1 are equal. Let show that for ν =0. From the asymptotics 47) and from the RHP 5) we have 75) χ 0 x, λ) =χ 1 x, λ)g 1 λ), G 1 λ) =Ŝ+ 1 λ)s 1 λ), λ l 1, where G 1 λ) belongs to an SL2) subgroup of SL) and is such that it commutes with Θ 1 x x ). Thus we conclude that 76) R 1 x, x,λe +i0 )=R 1 x, x,λe i0 ), λ l 1. Analogously we prove that R ν x, x,λe +i0 ) has no jumps on the other rays l ν. The jumps on the rays b ν appear because of two reasons: first, because of the functions Θ ν x x ) and second, it is easy to check that for λ b ν the kernel R ν x, x,λ) oscillates for x, x tending to ±. Thus on these lines the resolvent is unbounded integral operator. 2. Assume that λ Υ ν and consider the asymptotic behavior of R ν x, x,λ) for x, x. From eqs. 47) we find that 77) n Rijx, ν x,λ)= Xipx, ν λ)e iλjpx x ) Θ ν;pp x x ) ˆX pjx ν,λ). p=1 Due to the fact that χ ν x, λ) has the special triangular asymptotics for x and λ Υ ν and for the correct choice of Θ ν x x ) 72) we check that the right hand side of 77) falls off exponentially for x and arbitrary choice of x. All other possibilities are treated analogously.. For λ b ν the arguments of 2) can not be applied because the exponentials in the right hand side of 77) Im λ =0only oscillate. Thus we conclude that R ν x, x,λ) for λ b ν is only a bounded function and thus the corresponding operator Rλ) is an unbounded integral operator. 4. The proof of eq. 74) follows from the fact that Lλ)χ ν x, λ) =0and 78) Θx x ) x = 1δx x ). Lemma 1. The poles of R ν x, x,λ) coincide with the poles of the dressing factors ux, λ) and its inverse u 1 x, λ).

17 ON KAUP-KUPERSHCHMIDT TYPE EQUATIONS AND THEIR SOLITON SOLUTIONS 17 Proof 2. The proof follows immediately from the definition of R ν x, x,λ) and from Remark. Thus we have established that dressing by the factor ux, λ), we in fact add to the discrete spectrum of the Lax operator 6N 1 +12N 2 discrete eigenvalues; for N 1 = N 2 =1 they are shown on fig Discussion and conclusions On the example of the KKE we analyzed the relation between the scalar ordinary differential operators and the Kac-Moody algebras. Using the dressing method we established that KKE and its gauge equivalent MKdV have two types of one-soliton solutions, which generically are not traveling wave solutions. The dressing method adds discrete eigenvalues to the spectrum of L which comes in sextuplets for each soliton of first type and in dodecaplets 12-plets) for the solitons of second type. Still open is the question of constructing regular soliton solutions and to study the properties of the generic one soliton solutions that are. not traveling waves. We have constructed the FAS of L which satisfy a RHP on the set of rays l ν. We also constructed the resolvent of the Lax operator and proved that its continuous spectrum fills up the rays b ν rather than l ν. From fig. 1 we see that the eigenvalues corresponding to the solitons of first type lay on the continuous spectrum of L. This explains why the solitons of first type are singular functions. Using the explicit form of the resolvent R ν x, x,λ) and the contour integration method one can derive the completeness relation of the FAS. As a further development we note, that one can use the expansions over the squared solutions [0] to derive the action-angle variables of the NLEE in the hierarchy. These expansions are, in fact, spectral decompositions of the relevant recursion operators Λ k which in addition possess important geometrical properties [1-]. The author is grateful to Dr. Alexander Stefanov and to Prof. R. Ivanov for useful discussions and help in preparing the manuscript. Appendix A. The basis of A 1) 2 The basis of A 1) 2 is obtained from the Cartan-Weyl basis of A 2 15) by taking the average with the Coxeter automorphism. In this case the Coxeter automorphism is represented by CX C 1, C =

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