Symbolic Computation of. Conserved Densities and Fluxes for. Nonlinear Systems of Differential-Difference. Equations

Transcription

1 Symbolic Computation of Conserved Densities and Fluxes for Nonlinear Systems of Differential-Difference Equations by Holly Eklund

2 Copyright c 003 Holly Eklund, All rights reserved. ii

3 A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Masters of Science (Mathematical and Computer Sciences). Golden, Colorado Date Signed: Holly Eklund Approved: Dr. Willy Hereman Mathematical and Computer Sciences Thesis Advisor Golden, Colorado Date iii Dr. Graeme Fairweather Head Department Mathematical and Computer Sciences

4 ABSTRACT Two algorithms are presented to find conserved densities and fluxes of nonlinear systems of differential-difference equations. Both algorithms utilize the scaling properties of lattice equations to reduce the problem to a calculus and linear algebra problem. The two algorithms are illustrated for the Kac-van Moerbeke, Toda, and Ablowitz-Ladik lattices. The first method leads to a three step algorithm which utilizes the dilation invariance of the conservation laws to construct the form of the density. For this method, the discrete Euler operator or discrete variational derivative is an advantageous tool. The algorithm is implemented in Mathematica. The package is called DDEDensityFlux.m. The key applications are to analyze the discretizations of the Korteweg-de Vries (KdV), and modified Korteweg-de Vries (mkdv) lattices. A combination of the KdV and mkdv lattices is also considered. The second method leads to a five step algorithm that is primarily useful in determining fluxes. Both of the algorithms presented could be used to investigate the integrability of semi-discrete lattices. iv

5 TABLE OF CONTENTS ABSTRACT iv LIST OF TABLES vii ACKNOWLEDGMENTS viii Chapter 1 INTRODUCTION Chapter DIFFERENTIAL-DIFFERENCE EQUATIONS Definitions Conservation laws Example: The Kac-van Moerbeke lattice Dilation Invariance: The concept behind our algorithms Density-flux pairs Tool 1: Equivalence criterion Tool : The discrete Euler operator (variational derivative) Chapter 3 THE FIRST MATHEMATICAL METHOD AND AL- GORITHM The method Steps of the algorithm Chapter 4 IMPLEMENTATION AND SOFTWARE Existing software The new code DDEDensityFlux.m Using the software: A sample session Options for the user Implementation issues Chapter 5 COMPUTATION OF CONSERVATION LAWS The Kac-van Moerbeke lattice The Toda lattice The Ablowitz-Ladik lattice v

6 5.3.1 Introducing a first weighted parameter (α) Introducing a second weighted parameter (β) Introducing Shifts Chapter 6 A SECOND METHOD TO DETERMINE DENSITIES AND FLUXES The second algorithm The Kac-van Moerbeke lattice The Toda lattice The Ablowitz-Ladik lattice Chapter 7 APPLICATIONS Discretization of the combined KdV-mKdV equation Discretization of the Korteweg-de Vries equation Discretization of the modified Korteweg-de Vries equation Chapter 8 CONCLUSION REFERENCES Appendix A DATA FILES vi

7 LIST OF TABLES 5.1 Ranks of terms in density candidate (R=3) for the KvM lattice Ranks of terms in density candidate (R=3) for the Toda lattice Ranks of terms in density candidate (R= 1 ) for the AL lattice Ranks of terms in density candidate (R= 1 ) for the AL lattice Ranks of terms in density candidate (R= 1 ) for the combined equation Ranks of terms in density candidate (R= 3 ) for the combined equation Ranks of terms in density candidate (R=1) for the combined equation Ranks of terms in density candidate (R= 1 ) for the combined equation Ranks of terms in density candidate (R= 5 ) for the combined equation Ranks of terms in density candidate (R= 3 ) for the combined equation Ranks of terms in density candidate (R=1) for the KdV discretization Ranks of terms in density candidate (R=1) for the KdV discretization Ranks of terms in density candidate (R= 1 ) for the mkdv discretization Ranks of terms in density candidate (R=1) for the mkdv discretization Ranks of terms in density candidate (R= 1 ) for the mkdv discretization 99 vii

8 ACKNOWLEDGMENTS I greatly appreciate the help of Dr. Willy Hereman, my thesis advisor for his clear explanations, helpful input, and patience throughout. Without the support of the National Science Foundation Computer Science, Engineering and Mathematics Scholarships program, and the National Science Foundation research award CCR , this research would not have been possible. Many thanks go to my thesis committee members, Dr. Barbara Moskal and Dr. Paul Martin for all of their comments and suggestions. Finally, my sincere gratitude goes to my fiancé Jeffrey Bellman and my family for their support and encouragement throughout. viii

9 1 Chapter 1 INTRODUCTION Differential-difference equations (DDEs) have been the focus of many nonlinear studies since the original work of Fermi, Pasta, and Ulam in the nineteen sixties [9]. Nonlinear DDEs describe many interesting physical phenomena including vibrations of particles in lattices and currents in electrical networks, Langmuir waves, and interactions between competing populations. Also, DDEs play an important role in queuing problems and discretizations in solid state physics and quantum fields. Lastly, they are used in numerical simulations of nonlinear PDEs [1]. Recently, there has been a renewed interest in DDEs (see e.g. [40] for a review of the literature). DDEs are semi-discretized as the single space variable is discretized, and time is kept continuous. This is in contrast to their fully discretized counterparts, called difference equations, in which there currently is also a great deal of interest (see e.g. [7, 1, 7, 8]). In this thesis we focus on one aspect of the integrability of DDEs, namely the computation of polynomial conserved densities and associated fluxes via direct methods which can be implemented in computer algebra systems. The first few conservation laws of a DDE may have a physical meaning, for instance conserved momentum and energy. Additional ones may facilitate the study of both quantitative and qualitative properties of solutions [3]. Also, the existence of a sequence of conserved densities predicts integrability of DDEs [13]. However, the absence of conserved densities does not preclude integrability. The integrable DDEs could indeed be disguised with a co-

10 ordinate transformation so that the transformed equation no longer admits conserved densities of polynomial type. The main result of this thesis is two-fold. First, I added several new aspects to the previously existing method used in condens.m [10], InvariantsSymmetries.m [1] and diffdens.m [11]. Second, I implemented the new components in Mathematica. The new package DDEDensityFlux.m is much more reliable than previous versions, and it calculates conservation laws for new, more complicated systems of DDEs. In this thesis, I describe two novel methods [0] to construct families of conserved densities and apply them to specific examples. (1) The first method relies heavily on the notion of dilation invariance. It is shown in this thesis that conservation laws for DDEs are dilation invariant. We utilize this fact and construct the form of the density, ρ n. Then, we may either use a shifting technique or the discrete Euler operator to find the unknown constants. Using the shifting technique to determine the constants, gives the flux, J n, automatically. Using the discrete Euler operator to find the unknown constants simplifies the calculation of the flux, J n, but does not determine it directly. The first method leads to the first algorithm which was implemented in Mathematica as DDEDensityFlux.m. () The second method, suggested by Hickman [0], is more theoretical and is primarily useful in determining fluxes. It leads to a five step algorithm that uses repeated decomposition of the identity operator I to find pieces in and outside the image (Im) of the operator = D I, where D is the up-shift operator. First, we determine the furthest negative shifted variable in expression E (i.e. D t ρ n after replacement from the system). The expression E is then split into two parts, A (j) and A (j+1), where A (j) contains all terms that are independent of the lowest shifted variable

11 3 and A (j+1) has terms dependent on the lowest shifted variable. We then down-shift A (j) and add D 1 A (j) to A (j+1). This process is repeated until D 1 A (j) + A (j+1) = 0. The constraint determines the unknown coefficients leading to the final form of the density, ρ n, and the flux, J n. The techniques described in this thesis are applicable to complicated nonlinear systems of DDEs. Yet, to keep the ideas transparent and avoid lengthy computation (which are best performed with Maple, Mathematica, or mupad) we use the Kac-van Moerbeke (KvM), Toda, and Ablowitz-Ladik (AL) lattices to illustrate our methods. Like the algorithms and Mathematica codes (for densities and generalized symmetries) in [14, 15, 17, 19], the methods in the present paper are restricted to polynomial densities and fluxes. There is a vast body of work on DDEs, including investigations of integrability criteria via the computation of densities, generalized and master symmetries, recursion operators, etc. For nonlinear DDEs several solution methods and integrability tests are applicable. The solution methods include symmetry reduction [9], and an extension of the spectral transform method [6]. Adaptations of the singularity confinement approach [33], the Wahlquist-Estabrook method [8], and the master symmetry technique [6] allow one to test integrability of DDEs. The most comprehensive integrability study of DDEs was done by Levi and colleagues [6, 30], Yamilov [46, 47] and co-workers [4, 5, 6, 36, 37, 38]. Their papers provide a classification of semi-discrete equations possessing infinitely many local conservation laws. Using the formal symmetry approach, they derive the necessary and sufficient conditions for the existence of local conservation laws, and provide an algorithm to construct them. In contrast to these algorithms, in this thesis we present new direct algorithms that allow one to compute conserved densities and fluxes of DDEs. Our algorithms are

12 4 fairly straightforward. They only rely on algebra, calculus, and a tool from variational calculus. Therefore, they can be implemented in computer algebra languages. The first algorithm is implemented in Mathematica as DDEDensityFlux.m. DDEDensityFlux.m is based on previously existing methods used in condens.m, InvariantsSymmetries.m, and diffdens.m. The program condens.m automatically carries out the lengthy symbolic computations for the construction of conserved densities [13]. The package InvariantsSymmetries.m is a straightforward algorithm for the symbolic computation of generalized (higher-order) symmetries of nonlinear evolution equations and lattice equations [15, 16]. The code diffdens.m calculates conserved densities for several well-known lattice equations [17, 14]. The thesis is organized as follows. Chapter covers preliminary material about conservation laws of DDEs. Here we utilize equivalence criteria to simplify our calculations. An analogy to a result for PDEs, the discrete Euler operator (or variational derivative) is introduced as a valuable tool to test conserved densities. We prove the necessary and sufficient condition for a function of a discrete variable (and its shifts) to be the total difference of another function of discrete variables. A few simple examples illustrate the concepts. Chapter 3 describes the first algorithm for determining densities and fluxes. The first algorithm is organized into three steps. In Chapter 4, we address existing related algorithms. In addition, we explain how to use the developed software called DDEDensityFlux.m. Chapter 4 also includes a description of how the computation of the fluxes and the discrete Euler operator were implemented. The KvM, Toda, and AL lattices are used as examples to illustrate the three step algorithm in Chapter 5. The importance of introducing shifts in the density

13 5 (specifically for complicated examples) is demonstrated for the AL lattice. The chapter concludes with applications of the Euler operator to find densities and fluxes of the KvM, Toda, and AL lattices. In Chapter 6 we focus on the computation of the associated fluxes. We illustrate how the second method could be used to compute densities as well as fluxes. We revisit the KvM, Toda, and AL lattices to illustrate the method. We show some additional applications of the first algorithm in Chapter 7. Here we use our methods to compute densities and fluxes for discretizations of the Kortewegde Vries (KdV) and modified Korteweg-de Vries (mkdv) equations, as well as a combined equation. We draw some conclusions in Chapter 8. In summary, the original contribution of this thesis is two-fold: improved algorithms to determine conserved densities and fluxes of DDEs, and their implementation in Mathematica. We offer the scientific community a symbolic package called DDE- DensityFlux.m that carries out the tedious calculations of conserved densities and fluxes of nonlinear DDEs. The software and data files are available from various sources.

14 6 Chapter DIFFERENTIAL-DIFFERENCE EQUATIONS In this chapter we review preliminary material [0] about densities and fluxes of nonlinear DDEs. We also show a few simple examples..1 Definitions Definition.1 Differential-difference equations (DDEs) are equations that are continuous in time and discretized in space. They are of the following form: u n = f(u n l, u n l+1,..., u n,..., u n+m 1, u n+m ) (.1) with f u n l f u n+m 0, where n is an arbitrary integer. In general, f is a nonlinear vector-valued function of a finite number of dynamical variables, each u n is a vector-valued function of t, and u n is the usual time derivative of u n (t). The index n may lie in Z, or the u n+k may be periodic, u n+k = u n+k+n. The integer l is the furthest negative shift, and m is the furthest positive shift of any variable in (.1). If l = m = 0 then the equation is local and reduces to a system of ordinary differential equations.

15 7 There are two notations for DDEs. The index n, may be omitted identifying u n u 0, u n±p u ±p, etc. However, for clarity we commit to including the index n. Definition. The up-shift operator D is defined by D u n+k = u n+k+1. (.) Definition.3 Its inverse, the down-shift operator, is given by D 1 u n+k = u n+k 1. (.3) Thus we have u n+k = D k u n. The action of D and D 1 is extended to functions by acting on their arguments. For example, D g(u n p, u n p+1,..., u n+q ) = g(d u n p, D u n p+1,..., D u n+q ) = g(u n p+1, u n p+,..., u n+q+1 ). In particular, we have ( ) D u g(u n p, u n p+1,..., u n+q ) = n+k u g(u n p+1, u n p+,..., u n+q+1 ). n+k+1 In order to standardize notation, we consider p and q to be positive integers. Identify p with the furthest negative shift of any variable in the system, and q with the furthest positive shift of any variable in the system. Moreover, for equations of type (.1), the shift operator commutes with the time derivative ( ) D ddt u n = D f(u n l, u n l+1,..., u n,..., u n+m 1, u n+m ) = f(u n l+1, u n l+,..., u n+1,..., u n+m, u n+m+1 )

16 8 = d dt u n+1 = d dt (D u n). Definition.4 Next, we define the (forward) difference operator, = D I, by u n+k = (D I) u n+k = u n+k+1 u n+k, where I is the identity operator. This operator takes the role of a spatial derivative on the shifted variables as many examples of DDEs arise from the discretizations of a PDE in (1 + 1) variables [35]. The difference operator extends to functions and we have g(u n p, u n p+1,..., u n+q ) = (D I) [g(u n p, u n p+1,..., u n+q )] = D [g(u n p, u n p+1,..., u n+q )] I [g(u n p, u n p+1,..., u n+q )] = g(u n p+1, u n p+,..., u n+q+1 ) g(u n p, u n p+1,..., u n+q ). Definition.5 For any function g = g(u n p, u n p+1,..., u n+q ), the total time derivative D t g is computed as D t g(u n p, u n p+1,..., u n+q ) = ( g u ) u n p + + ( g n p u ) u n + n +( g u ) u n+q n+q = ( g u )D p u n + + ( g n p u )D 0 u n + + ( g n u )D q u n n+q q = ( g u D k ) u n n+k = ( k= p q k= p g u n+k D k )f(u n l, u n l+1,..., u n+m 1, u n+m ) (.4)

17 9 on solutions of (.1). A simple calculation shows that the shift operator D commutes with D t. Note that we consider only autonomous functions and systems, i.e f and g do not explicitly depend on t. Hence, f t = 0 and g t = 0.. Conservation laws Definition.6 A function ρ n = ρ n (u n p, u n p+1,..., u n+q ) is a (conserved) density of (.1) if there exists a function J n = J n (u n r, u n r+1,..., u n+s ), called the (associated) flux, such that D t ρ n + J n = 0 (.5) or equivalently, D t ρ n = J n = (D I)J n = [J n J n+1 ] is satisfied on the solutions of (.1). (.5) is called a local conservation law. Any shift of a density, D k ρ n, is trivially a density since D t D k ρ n + D k J n = D k (D t ρ n + J n ) = D t 0 = 0. The associated flux is D k J n.

18 10 Constants of motion for (.1) are easily obtained from densities and their shifts. Indeed, for any density ρ n with corresponding flux J n, consider The total time derivative of Ω is q Ω = D k ρ n. (.6) k= p q D t Ω = D k J n = k= p q k= p ( D k+1 D k) J n = ( D q+1 D p) J n. Applying appropriate boundary conditions, for example, lim u n = 0, lim u n+p = 0, (.7) n n one gets the conservation law D t k= D k ρ n = lim D q+1 J n + lim D p J n = 0. q p For a periodic chain, where u k = u k+n, after summing over a period, one obtains ( N ) D t D k ρ n = D N+1 J n + D 0 J n = J n + J n = 0. k=0 In either case, Ω is a constant of motion of (.1) since Ω does not change with time.

19 11 Definition.7 A density which is a total difference, ρ n = F n, (.8) (so that D t ρ n = D t F n and therefore J n = D t F n is an associated flux), is called trivial. These densities lead to trivial conservation laws since q Ω = D k F n = D q+1 F n D p F n k= p holds identically, not just on solutions of (.1)..3 Example: The Kac-van Moerbeke lattice Example.1 Consider the semi-discrete KvM lattice u n = u n (u n+1 u n 1 ), (.9) where as usual, u n = du n /dt. Note that (.9) is equivalent to the notationally simpler u 0 = u 0 (u 1 u 1 ). However, as previously mentioned, for the sake of illustration we shall commit to the first standard notation. Eq. (.9) is often referred to as a Volterra lattice [45, 47], although it is a special case of the two-component Volterra system [, 36].

20 1 It arises in the study of Langmuir oscillations in plasmas, population dynamics, quantum field theory, polymer science, and appears in the context of matrix factorization (see references in [40]). Eq. (.9) appears in the literature in other forms, including Ṙ n = 1 (e R n 1 e R n+1 ), (.10) and ẇ n = w n (w n+1 w n 1), (.11) which relate to (.9) by simple transformations [40]. However, (.9) most conveniently illustrates this algorithm. It has the following pairs of conserved densities and fluxes [17] of rank 3: ρ (1) n = u n, J (1) n = u n 1 u n, (.1) ρ () n = 1 u n + u n u n+1, (.13) J () n = u n 1 u n u n 1 u n u n+1, (.14) ρ (3) n = 1 3 u n 3 + u n u n+1 (u n + u n+1 + u n+ ), (.15) J (3) n = (u n 1 u n 3 + u n 1 u n u n+1 + u n 1 u n u n+1 + u n 1 u n u n+1 u n+ ). (.16) Then clearly (.5) holds since D t ρ (1) n = u n = u n (u n+1 u n 1 ) = [J n (1) J n+1] (1) = J n (1) and D t ρ () n = u n u n + u n u n+1 + u n u n+1

21 13 = u n (u n+1 u n 1 ) + u n u n+1 (u n+1 u n 1 ) + u n u n+1 (u n+ u n ) = [J () n J () n+1] = J () n. This holds for all density flux pairs of (.1)..4 Dilation Invariance: The concept behind our algorithms Many definitions and key ideas are given here. These concepts will be used repeatedly throughout this thesis. Definition.8 Key to our methods is the concept of dilation symmetry or (scaling) (t, u, v) (λ a t, λ b u, λ c v), (.17) where λ is an arbitrary parameter. Note that a, b, and c could be fractions. If a 0, without loss of generality we may set w( d ) = 1. Therefore, a = 1 and u corresponds to b derivatives with respect to t. dt We denote this by u db dt b, and v corresponds to c derivatives with respect to t, or v dc dt c. The terms on the right hand side of (.9) both have rank R =, since each of these (monomial) terms is quadratic [4]. Recall the KvM lattice given by (.9). The equation is invariant under a dilation (scaling) symmetry. Indeed, (.9) is invariant under (t, u n ) (λ 1 t, λu n ). Therefore, u n corresponds to one derivative with respect to t. We denote this u n d/dt. We say the weight of u n is one. By choice, we set w(t) = 1 or w( d dt ) = 1. Definition.9 The weight, w, of a variable is equal to the number of derivatives with respect to t the variable carries.

22 14 Weights of dependent variables and parameters are non-negative, rational, and independent of n. Note that weights may be fractions. The weight of each term in a single equation is equal to the weights of the other terms in the same equation. It is always legitimate to consider w(u n ) 0 and w(v n ) 0, since zero weights would lead to a trivial case. Definition.10 The rank of a monomial is defined as the total weight of the monomial. Once an equation is made scaling invariant, all the terms (monomials) in a particular equation have the same rank. This property is called uniformity in rank. A system is uniform in rank if every equation is uniform in rank. Note that the ranks of the various equations may differ from each other. By definition, (.9) is uniform in rank and the rank is. Conservation laws are also uniform in rank. This property will be addressed in further detail shortly. Definition.11 An equation is scaling invariant iff weights can be assigned to each term in the equation so that the total weights of all the terms are equal. A system of DDEs is scaling invariant is every equation is scaling invariant. If a system is scaling invariant, there is a consistent system of equations for the unknown weights. So, we may assign the appropriate resulting weights to the variables. Note that different equations in the vector equation (.1) may have different ranks. For more complicated DDEs, it is also convenient to consider the case when w( d dt ) = 0, i.e. a = 0. Using this extra scale lets us to group terms in ρ n according to w( d ) = 0. This allows for much simpler calculations because for any given density dt that is uniform in rank under Scale 1, we may separate it into pieces which are uniform in rank under Scale 0. We now give the formal definitions of the two scales.

23 15 Definition.1 When computing the weights in a system of DDEs, one may choose w( d ) = 1, and calculate the weights of the variables accordingly. We will call this dt Scale 1. In other words, Scale 1 corresponds to the choice w( d dt ) = 1. Definition.13 When computing the weights in a system of DDEs, one may choose w( d ) = 0, and calculate the weights of the variables accordingly. We will call this dt Scale 0. In other words, Scale 0 corresponds to the choice w( d dt ) = 0. For systems that are not scaling invariant, we use the following trick: We introduce one (or more) auxiliary parameter(s), and treat them as dependent variables with an appropriate scaling. When introducing the parameter(s), it is vital to assume that no parameter equals 0. Setting parameters zero would make some terms vanish and this would alter the problem entirely. By introducing auxiliary parameters and assigning weights to them, we can make each term in a single equation of the same weight as the other terms. This process creates a modified, but scaling invariant system of DDEs. Furthermore, by extending the action of the dilation symmetry to the space of independent and dependent variables and parameters, we are able to apply our first algorithm to any polynomial system of DDEs. However, this comes at a great cost because by introducing such auxiliary parameter(s), the resulting ρ n and J n are no longer linearly independent..5 Density-flux pairs Example. Recall the density flux pair in (.15) and (.16), namely: ρ (3) n = 1 3 u n 3 + u n u n+1 (u n + u n+1 + u n+ ), (.18) J (3) n = (u n 1 u n 3 + u n 1 u n u n+1 + u n 1 u n u n+1 + u n 1 u n u n+1 u n+ ). (.19)

24 16 We can verify that (.18) and (.19) indeed obey D t ρ (3) n = [J (3) n J (3) n+1] since D t ( 1 3 u n 3 + u n u n+1 (u n + u n+1 + u n+ )) = u n u n + u n u n u n+1 + u n u n+1 + u n u n+1 + u n u n+1 u n+1 + u n u n+1 u n+ +u n u n+1 u n+ + u n u n+1 u n+ = u n 1 u n 3 u n 1 u n u n+1 + u n u n+1 3 u n 1 u n u n+1 + u n u n+1 u n+ u n 1 u n u n+1 u n+ + u n u n+1 u n+ + u n u n+1 u n+ u n+3 = (u n 1 u n 3 + u n 1 u n u n+1 + u n 1 u n u n+1 + u n 1 u n u n+1 u n+ ) +(u n u n u n u n+1 u n+ + u n u n+1 u n+ + u n u n+1 u n+ u n+3 ) = [J (3) n J (3) n+1]. In general, for any conservation law D t ρ n + J n = 0, rank(j (3) n ) = w(d t ) + rank(ρ (3) n ) = 1 + rank(ρ (3) n ). The reason for the uniformity in rank of the conservation law is obvious: In computing D t ρ n we use the scaling invariant DDE to replace all time derivatives. In doing so, the conservation law inherits the scaling symmetry of the given DDE. Therefore, ρ n, J n, and the terms in the conservation law must all be uniform in rank. So, we will use the fact that (.5) is uniform in rank when computing conserved densities

25 17 and fluxes. Indeed, in Step of our first algorithm we will build a candidate ρ n as a linear combination of monomials of a given rank (say R). Example.3 Now consider the Toda lattice [18, 41] ÿ n = exp (y n 1 y n ) exp (y n y n+1 ). (.0) In (.0), y n is the displacement from equilibrium of the nth particle with unit mass under an exponential decaying interaction force between nearest neighbors. With the change of variables, u n = ẏ n, v n = exp (y n y n+1 ), lattice (.0) can be written in algebraic form u n = v n 1 v n, v n = v n (u n u n+1 ). (.1) We can compute a couple of conservation laws for (.1) by hand. Indeed, u n = D t ρ n = v n 1 v n = [J n J n+1 ] with J n = v n 1. We denote this first pair by ρ (1) n = u n, J (1) n = v n 1.

26 18 After some work, we obtain a second pair: ρ () n = 1 u n + v n, J () n = u n v n 1. Key to our method is the observation that (.5) and (.1), together with the above densities and fluxes, are invariant under the dilation symmetry (t, u n, v n ) (λ 1 t, λu n, λ v n ), (.) where λ is an arbitrary parameter. The result of this dimensional analysis can be stated as follows: u n corresponds to one derivative with respect to t; for short, u n d dt. Similarly, v n d dt. Scaling invariance, which is a special Lie-point symmetry, is an intrinsic property of many integrable nonlinear DDEs. For scaling invariant systems such as (.9) and (.1), it suffices to consider the dilation symmetry on the space of independent and dependent variables. Certainly, we may verify D t ρ () n = [J () n J () n+1], since D t [ 1u n + v n ] = u n u n + v n = u n (v n 1 v n ) + v n (u n u n+1 ) = u n v n 1 u n+1 v n = [J () n J n+1 () ].

27 19.6 Tool 1: Equivalence criterion Definition.14 Two densities ρ n, ρ n are called equivalent if ρ n ρ n Im, i.e. ρ n ρ n = F n for some F n. Equivalent densities, denoted as ρ n ρ n, yield the same conservation law. Note that (.8) expresses that ρ n 0. It is easy to verify that compositions of D and D 1 define an equivalence relation on monomials. The equivalence criterion will be used in Step 3 of our first algorithm. Definition.15 In the algorithms in this thesis, we will use the following equivalence criterion: if two monomials, m 1 and m, are equivalent, m 1 m, then m 1 m = M n for some polynomial M n that depends on u n and its shifts. For example, m 1 = u n u n u n 1 u n+1 = m since u n u n u n 1 u n+1 = M n = M n+1 M n = u n 1 u n+1 ( u n u n ), (.3) with M n = u n u n. Definition.16 We call the main representative of an equivalence class, the monomial of that class with n as lowest label on u (or v). For example, u n u n+ is the main representative of the class with elements u n u n, u n 1 u n+1, u n+1 u n+3, etc. We use lexicographical ordering to resolve conflicts. That is, u n v n+ (not u n v n ) is the main representative in the class with elements u n 3 v n 1, u n+ v n+4, etc.

28 0 Simply stated, all shifted monomials are equivalent. For example, u n 1 v n+1 u n+ v n+4 u n 3 v n 1, with u n v n+ as the main representative. This equivalence relation holds for any function of the dependent variables, but for the construction of conserved densities we will apply it only to monomials..7 Tool : The discrete Euler operator (variational derivative) Completely integrable PDEs and DDEs exhibit unique analytic properties. For instance, most completely integrable differential equations have infinitely many symmetries and conserved densities. For PDEs, in the computation of ρ n and J n, one has to determine whether or not E = D t ρ n can be written as D x J n. To verify whether or not J n exists, one can use a tool from variational calculus. It is well-known that a (continuous) function g(u(x), u (x), u (x),..., u n (x)) can be integrated with respect to x if and only if (iff) the variational derivative, L u, of g vanishes. Formally, g = D x h for some function h(u(x), u (x), u (x),..., u m (x)) iff L u (g) = 0. Here, D x refers to total differentiation with respect to x and L u is the continuous Euler operator (variational derivative). In particular, a function is a conserved density of a PDE iff its time-derivative is in the kernel (Ker) of the Euler operator (see e.g. [13]). We now present a discrete analog of this important result. One can test for complete integrability by examining conservation laws of DDEs. Namely, E = D t ρ n = J n, where ρ n is the density and J n is the flux. If E = D t ρ n is a total difference, then E = D t ρ n can be written as [J n J n+1 ], and if E = D t ρ n is not a total difference, then the nonzero terms must vanish identically. The discrete Euler operator (discrete variational derivative) is one tool to test if a discrete expression is a total difference. To verify whether or not E is a total difference we will now introduce the discrete analog of the Euler operator, denoted L un.

29 1 Definition.17 The discrete Euler operator or discrete variational derivative, L u, is identified with the following equation: q L un (g) = D k k= p u g n+k = D p u g + D p 1 g n p u D 0 g n p+1 u D q g n u.(.4) n+q Note that we can rewrite the Euler operator as L un (g) = u n q k= p D k g, (.5) and that q k= p D k = D q+1 D p. If we know that ρ n is a density, then E = D t ρ n is a total difference because E = D t ρ n = J n, where J n is the flux associated with ρ n. In summary, if L un (E) = 0, then E = D t ρ n equals J n for some J n, and vice versa. So, the Euler operator is most useful in our first algorithm. It will be utilized in Step 3 of our first algorithm. Example.4 Recall the semi-discrete KvM lattice identified by (.9). It is well known that ρ (3) n = 1 3 u n 3 + u n u n+1 (u n + u n+1 + u n+ ), (.6) J (3) n = (u n 1 u n 3 + u n 1 u n u n+1 + u n 1 u n u n+1 + u n 1 u n u n+1 u n+ ) (.7) are a density-flux pair for (.9). If we replace the correct numerical constants by

30 arbitrary constants, then ρ (3) n = c 1 u n 3 + c u n u n+1 + c 3 u n u n+1 + c 4 u n u n+1 u n+ (.8) is the form of the density candidate. We compute D t ρ (3) n = 3c 1 u n u n + c u n u n+1 + c u n u n+1 u n + c 3 u n u n+1 u n+1 +c 3 u n+1 u n + c 4 u n u n+1 u n+ + c 4 u n u n+ u n+1 + c 4 u n+1 u n+ u n.(.9) After replacing the time derivatives (using (.9)) we get E = D t ρ (3) n = 3c 1 u n 3 u n+1 3c 1 u n 1 u n 3 + c u n u n+1 u n+ c u n 3 u n+1 +c u n u n+1 c u n 1 u n u n+1 + c 3 u n u n+1 u n+ c 3 u n u n+1 +c 3 u n u n+1 3 c 3 u n 1 u n u n+1 + c 4 u n u n+1 u n+ u n+3 + c 4 u n u n+1 u n+ +c 4 u n u n+1 u n+ c 4 u n 1 u n u n+1 u n+. (.30) Applying the discrete Euler operator to E gives q L un (E) = D k k= p u E n+k = D p u E + D p 1 E n p u D 0 E n p+1 u D q E n u n+q = (9c 1 u n u n+1 3c u n u n+1 9c 1 u n 1 u n + c u n u n+1 u n+ c 4 u n u n+1 u n+ + 4c u n u n+1 4c 3 u n u n+1 4c u n 1 u n u n+1 +c 3 u n+1 u n+ + c 3 u n+1 3 c 3 u n 1 u n+1 + c 4 u n+1 u n+ u n+3 +c 4 u n+1 u n+ c 4 u n 1 u n+1 u n+ ) + D 1 (3c 1 u n 3 c u n 3 + c u n u n+ c 4 u n u n+ + 4c u n u n+1 4c 3 u n u n+1 c u n 1 u n + 4c 3 u n u n+1 u n+

31 3 +3c 3 u n u n+1 c 3 u n 1 u n u n+1 + c 4 u n u n+ u n+3 + c 4 u n u n+ c 4 u n 1 u n u n+ ) + D (c u n u n+1 c 4 u n u n+1 + c 3 u n u n+1 +c 4 u n u n+1 u n+3 + c 4 u n u n+1 u n+ c 4 u n 1 u n u n+1 ) + D 3 (c 4 u n u n+1 u n+ ) +D( 3c 1 u 3 n c u n u n+1 c 3 u n u n+1 c 4 u n u n+1 u n+ ) = 9c 1 u n u n+1 3c u n u n+1 9c 1 u n 1 u n + c u n u n+1 u n+ c 4 u n u n+1 u n+ + 4c u n u n+1 4c 3 u n u n+1 4c u n 1 u n u n+1 +c 3 u n+1 u n+ + c 3 u n+1 3 c 3 u n 1 u n+1 + c 4 u n+1 u n+ u n+3 + c 4 u n+1 u n+ c 4 u n 1 u n+1 u n+ + 3c 1 u n 1 3 c u n c u n 1 u n+1 c 4 u n 1 u n+1 +4c u n 1 u n 4c 3 u n 1 u n c u n u n 1 + 4c 3 u n 1 u n u n+1 + 3c 3 u n 1 u n c 3 u n u n 1 u n + c 4 u n 1 u n+1 u n+ + c 4 u n 1 u n+1 c 4 u n u n 1 u n+1 +c u n u n 1 c 4 u n u n 1 + c 3 u n u n 1 + c 4 u n u n 1 u n+1 +c 4 u n u n 1 u n c 4 u n 3 u n u n 1 + c 4 u n 3 u n u n 1 3c 1 u n+1 3 c u n+1 u n+ c 3 u n+1 u n+ c 4 u n+1 u n+ u n+3. Grouping like terms gives L un (E) = (9c 1 3c )u n u n+1 + ( 9c 1 + 3c 3 )u n 1 u n + (c c 4 )u n u n+1 u n+ +(4c 4c 3 )u n u n+1 + ( 4c + 4c 3 )u n 1 u n u n+1 + (c 3 c )u n+1 u n+ +( 3c 1 + c 3 )u n ( c 3 + c 4 )u n 1 u n+1 + (c 4 c 4 )u n+1 u n+ u n+3 +(c 4 c 3 )u n+1 u n+ + ( c 4 + c 4 )u n 1 u n+1 u n+ + (3c 1 c )u n 1 3 +(c c 4 )u n 1 u n+1 + (4c 4c 3 )u n 1 u n + ( c + c 3 )u n u n 1 +( c 3 + c 4 )u n u n 1 u n + ( c 4 + c 4 )u n u n 1 u n+1 +(c c 4 )u n u n 1 + ( c 4 + c 4 )u n 3 u n u n 1. (.31)

32 4 So, ρ n will be a true density if E = D t ρ n is a total difference. Hence, L un (E) = 0, which leads to the system 3c 1 + c 3 = 0, c 3 c 4 = 0, 3c 1 c = 0, c c 4 = 0, c c 3 = 0. (.3) Solving this linear system gives 3c 1 =c =c 3 =c 4. If we let c 1 = 1, then c 3 =c 3 =c 4 =1. So, application of the discrete variational derivative to E = D t ρ n and setting the resulting expression equal to zero provides the linear system of c i. Hence, once the form of ρ n is known, we have reduced the problem of determining the constants to simple calculus and linear algebra.

33 5 Chapter 3 THE FIRST MATHEMATICAL METHOD AND ALGORITHM 3.1 The method The first algorithm requires solving the total-difference condition for the unknown density (ρ n ) and then constructing the associated flux (J n ). It can be described in three basic steps. First, it is necessary to determine the weights of the variables according to both Scale 1 and Scale 0. More formal definitions for Scale 1 and Scale 0 were discussed in Chapter. Then, we must construct the form of the density candidate. Finally, we determine the unknown coefficients in the density and the associated flux. Computation of the unknown coefficients can be done in two different ways. One may choose to compute the coefficients by using a shifting routine. Or, one may choose to compute the coefficients by employing the discrete Euler operator. 3. Steps of the algorithm Step 1: Determine the weights of the variables according to both Scales As mentioned previously in Chapter, (.1) is either scaling invariant or can be made scaling invariant by introducing auxiliary weighted parameters. Therefore, every monomial in (.1) has the same rank R. So, we get two different systems for the weights of the dependent variables. In Chapter, we used one scale, Scale 1, where t is replaced by λ 1 t. This scale corresponds to w( d ) = 1. One can also consider the dt

34 6 case where t is left unscaled, i.e. t is replaced by λ 0 t. Hence, w(t) = 0 or w( d ) = 0. dt We call this Scale 0. The multiple-scale approach was suggested by Sanders [34]. We may then solve the two systems for the weights of the dependent variables. Step : Construct the form of the density This step involves finding the building blocks (monomials) of a polynomial density with a prescribed rank R. Recall that all terms in the density must have the same rank R. Since we may introduce parameters with weights, the fact that the density is a sum of monomials of uniform rank does not necessarily imply that the density must be uniform in rank with respect to the dependent variables. Let V be the list of all the variables with positive weights, including the parameters with weight. The following procedure is used to determine the form of the density of rank R: Form the set G of all monomials of rank R or less by taking all appropriate combinations of different powers of the variables in V. For each monomial in G, introduce the appropriate number of derivatives with respect to t so that all the monomials exactly have weight R. Gather in set H all the terms that result from computing the various derivatives. Identify the monomials that belong to the same equivalence classes and replace them by the main representatives. Call the resulting simplified set I, which consists of the building blocks of the density with desired rank R. Linear combinations of the elements in I with constant coefficients, c i, gives the form of polynomial density of rank R.

35 7 Using the linear combinations, construct a table of the terms in ρ n based on Scale 1, where w( d ) = 1. Adjacent to the terms, record the rank of each term according to dt Scale 1 and also Scale 0. We may now group the terms by rank according to Scale 0. The new ρ n candidate is a linear combination of any one of these groups of terms with constants c i for i = 1,,...n. Note that for simple examples, using the multiple scale technique may not eliminate terms. Step 3: Determine the unknown coefficients The following shifting procedure simultaneously determines the constants, c i, and the form of the flux J n : Compute D t ρ n and use (.1) to remove all t derivatives. Once we have replaced from (.1) we represent D t ρ n as E. Prior to replacement from (.1), ρ n depends on u n, u n+1,..., u n+q 1, u n+q, so ρ n (u n, u n+1,..., u n+q 1, u n+q ), (3.1) where q is a positive integer as before, and q is furthest positive shift of any variable in ρ n. However, after replacement from (.1), D t ρ n = E depends on u n p, u n p+1,..., u n+ q 1, u n+ q, thus E(u n p, u n p+1,..., u n+ q 1, u n+ q ), (3.) where p and q are positive integers as before, and p is the furthest negative shift of any variable in E, and q is furthest positive shift of any variable in E. For simplicity of notation, we will replace p p and q q. So, p always refers to the furthest negative shift of any variable and q represents the furthest positive shift of any variable.

36 8 Use the equivalence criterion to modify E. The goal is to introduce the main representatives and to identify the terms that match the pattern [J n J n+1 ]. In order to do so, we may add and subtract up-shifted or down-shifted monomials repeatedly to E until all main representatives are introduced. For example, if E = c 1 u n 1 u n v n 1, then we may add and subtract up-shifted or down-shifted monomials to get E = c 1 u n u n+1 v n + [c 1 u n 1 u n v n 1 c 1 u n u n+1 v n ]. Clever grouping ensures that E matches the pattern [J n J n+1 ], where the flux is the first piece in the pattern [J n J n+1 ]. Definition 3.1 If there are terms in E that do not match the pattern [J n J n+1 ], then these terms form the obstruction. They must vanish identically (i.e. the coefficients for any combination of the components of u n and their shifts must be zero). We know from (.5) that E = D t ρ n = J n = [J n J n+1 ]. Setting the obstruction equal to zero leads to a linear system S in the unknowns, c i. If S has parameters, careful analysis leads to conditions on these parameters guaranteeing the existence of densities. See [13] for a description of this compatibility analysis. Note that if we artificially introduced parameters to ensure scaling invariance, then there may be freedom in many of the constant coefficients in ρ n. We set such arbitrary coefficients equal to 1 one at a time when determining the density. The first algorithm is implemented in Mathematica as DDEDensityFlux.m. A description of how the code was developed based on previous work is described in Chapter 4. We also explain how the software is used, and various implementation issues that were encountered.

37 9 Chapter 4 IMPLEMENTATION AND SOFTWARE As noted previously, the methods used in the new code DDEDensityFlux.m are based on some existing methods. These methods were already incorporated in condens.m, diffdens.m, and InvariantsSymmetries.m which were implemented in Mathematica. 4.1 Existing software An algorithm for the symbolic computation of polynomial conserved densities for systems of nonlinear evolution equations is presented in [13]. The algorithm is implemented in Mathematica and is called condens.m [10]. The code is tested on several well-known partial differential equations from soliton theory. For systems with parameters, condens.m can be used to determine the conditions on these parameters so that a sequence of conserved densities might exist. As with DDEs, the existence of a large number of conservation laws is a predictor for integrability of the system. A straightforward algorithm for the symbolic computation of generalized (higherorder) symmetries of nonlinear evolution equations and lattice equations is presented in [15] and [16]. The scaling properties of the evolution or lattice equations are used to determine the polynomial form of the generalized symmetries. The coefficients of the symmetry can be found by solving a linear system. The method applies to polynomial systems of PDEs of first-order in time and arbitrary order in one space variable. Likewise, lattices must be of first order in time but may involve arbitrary

38 30 shifts in the discretized space variable. The algorithm InvariantsSymmetries.m [1] is implemented in Mathematica and can be used to test the integrability of both nonlinear evolution equations and semi-discrete lattice equations. With InvariantsSymmetries.m, generalized symmetries are obtained for several well-known systems of evolution and lattice equations in [15] and [16]. For PDEs and lattices with parameters, the code allows one to determine the conditions on these parameters so that a sequence of generalized symmetries exists. The existence of a sequence of such symmetries is a predictor for integrability. The software package InvariantsSymmetries.m was used by Sakovich (Institute of Physics, National Academy of Sciences, Minsk, Belarus) and Tsuchida (Dept. of Physics, University of Tokyo, Tokyo, Japan) in the investigation of the integrability of coupled nonlinear Schrödinger equations. The new algorithm DDEDensityFlux.m is based on diffdens.m [11] described in [17] and [14]. The code diffdens.m computes conserved densities of nonlinear differential-difference equations. The code diffdens.m for conservation laws of DDEs was successfully used by Tsuchida, Ujino and Wadati in a study of integrable semi-discretizations of the coupled modified Korteweg- de Vries equations [4, 43] and the study of integrable semidiscretizations of the coupled nonlinear Schrödinger equations [44]. Although condens.m, InvariantsSymmetries.m, and diffdens.m are precursors to the new algorithm DDEDensityFlux.m, there are many notable improvements to the new code. The additions included in DDEDensityFlux.m make the software much more reliable and able to calculate conservation laws for a much broader class of DDEs.

39 31 4. The new code DDEDensityFlux.m New to DDEDensityFlux.m is the implementation of the direct computation of the flux. This is a great improvement to the package because now the software is completely self-testing. In fact, the code automatically verifies (.5). The implementation of the discrete Euler operator to solve for the unknown constants is also new. The previous method of solving for such constants involved a shifting technique that is also described in this thesis. Utilizing the discrete Euler operator gives a much more mathematical calculation of conserved densities. However, as mentioned earlier, the discrete Euler operator does not solve for the flux directly. In addition, earlier versions of this package only consider a single-scale approach. The multi-scale approach is an improvement since now we can simplify the density candidate by separating it into pieces that are uniform in rank according to Scale 0. Therefore, we can calculate densities and fluxes for much more complicated systems of DDEs. This version of the software includes implementation of a new way to form the density including shifts on the dependent variable. Computation of some conservation laws is greatly simplified by introducing shifts on the dependent variable when building up the density candidate. The new code allows for two different choices to determine the shifted dependent variables. The two ways to consider the shifts are explained more in detail in Introducing Shifts in Chapter 5. Finally, DDEDensityFlux.m has an improved linear systems analyzer. New to the code is the simplification of the linear system before it is analyzed. In other words, it eliminates any duplicate results for the c i and uses any c i equal to zero when analyzing the rest of the equations in the linear system. So, the new code may now

40 3 handle more complicated linear systems for the unknown constants, c i. So, the ideas used in condens.m, InvariantsSymmetries.m, and diffdens.m are utilized in our new software package DDEDensityFlux.m. Although some of the algorithm was previously developed and implemented in diffdens.m, the new version of DDEDensityFlux.m includes many important improvements. These improvements not only make the DDEDensityFlux.m more reliable than previous versions, but they also make it possible to compute conservation laws for much more complicated DDEs. Some of these more complex systems are investigated in Chapter 7 of this thesis. DDEDensityFlux.m is written and implemented in Mathematica. The program DDEDensityFlux.m and all data files are available via anonymous FTP from mines.edu. The login name is anonymous and the password is your address. The files are in the subdirectory pub/papers/math cs dept/software/ddedensityflux.m. The software is also available from the scientific section of Hereman s homepage with URL: home/whereman/. DDEDensityFlux.m automatically carries out the tedious calculations needed to determine conserved densities and fluxes for nonlinear systems of DDEs. The code is completely menu driven which makes it very easy to use. We now proceed with an example of how the software is used. 4.3 Using the software: A sample session Consider the example for determining ρ n and J n of the KvM lattice of rank R=3. In[1]:= << DDEDensityFlux.m Reads in the code << DDEDensityFlux.m, and produces the following menu: *** MENU INTERFACE *** (page: 1)

41 ) Kac-van Moerbeke Equation (d kdv.m) ) Modified KdV Equation (with parameter) (d mkdv.m) 3) Modified (quadratic) Volterra Equation (d molvol.m) 4) Ablowitz-Ladik Discretization of NLS Equation (d ablnl1.m) 5) Toda Lattice (d toda.m) 6) Standard Discretization of NLS Equation (d stdnls.m) 7) Herbst/Taha Discretization of KdV Equation (d diskdv.m) 8) Herbst/Taha Discretization of mkdv Equation (d dimkdv.m) 9) Herbst/Taha Discretization of combined KdV-mKdV Equation (d herbs1.m) 10) Self-Dual Network Equations (d dual.m) nn) Next Page tt) Your System qq) Exit Program Taking option nn shows the remaining data cases *** MENU INTERFACE *** (page: ) ) Parameterized Toda Lattice (d ptoda.m) 1) Generalized Toda Lattice-1 (d gtoda1.m) 13) Generalized Toda Lattice- (d gtoda.m) 14) Henon System (d henon.m) nn) Next Page

42 34 tt) Your System qq) Exit Program Selecting choice 1 in the menu, the code continues with **************************************************************** WELCOME TO THE MATHEMATICA PROGRAM by UNAL GOKTAS, WILLY HEREMAN, AND HOLLY EKLUND FOR THE COMPUTATION OF CONSERVED DENSITIES AND FLUXES. Version 3 released on January 10, 003 Copyright **************************************************************** Working with the data file for the Kac-van Moerbeke (or Volterra) Equation. Equation 1 of the system with 1 equation(s): (u 1,n ) = u 1,n ( (u 1, 1+n ) + u 1,1+n ) Note that u 1,n is representative of u n. If there were a u,n, it would represent of v n, and u 3,n would represent w n, etc. LINEAR SYSTEM FOR THE WEIGHTS CORRESPONDING TO w(d/dt)=0: {zeroweightu[1]== zeroweightu[]} SOLUTION OF THE SCALING EQUATIONS for w(d/dt)=0: {zeroweightu[1] 0} LINEAR SYSTEM FOR THE WEIGHTS CORRESPONDING TO w(d/dt)=1: