Symbolic Computation of Lax Pairs of Nonlinear Partial Difference Equations

SYMBOLIC COMPUTATION OF LAX PAIRS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS

by Terry J. Bridgman c Copyright by Terry J. Bridgman, 2018

All Rights Reserved 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 Doctor of Philosophy (Mathematical and Computer Sciences).

Golden, Colorado Date

Signed: Terry J. Bridgman

Signed: Dr. Willy Hereman Thesis Advisor

Golden, Colorado Date

Signed: Dr. Gregory Fasshauer Professor and Head Department of Applied Mathematics and Statistics

ii ABSTRACT

This thesis is primarily concerned with the symbolic computation of Lax pairs for nonlinear systems of partial difference equations (P∆Es) which are defined on a quadrilateral and consistent around a cube (CAC). A literature survey provides historical context for the results presented in this thesis. Particular attention is paid to the origins of integrable P∆Es which are central to this dissertation. Pioneering work of Ablowitz & Ladik as well as Hirota gave rise to nonlinear P∆Es as discretizations of completely integrable partial differential equations. Subsequent investigations by Nijhoff, Quispel & Capel and Adler, Bobenko & Suris provided a strong impetus to the modern and ongoing study of fully discrete integrable systems covered in this thesis. An algorithmic method due to Nijhoff and Bobenko & Suris to compute Lax pairs for scalar P∆Es is reviewed in detail. The extension and implementation of that algorithm for systems of P∆Es are part of the novel research in this thesis. The algorithm has been implemented in the syntax of Mathematica, a major and commonly used computer algebra system. A symbolic software package, LaxPairPartialDifferenceEquations.m accompanies the thesis. The code automatically (i) determines whether or not P∆Es have the CAC property, (ii) computes Lax pairs for nonlinear P∆Es that are CAC; and (iii) verifies if Lax pairs satisfy the Lax equation. Lax pairs are presented for the scalar integrable P∆Es classified by Adler, Bobenko, and Suris as well as for numerous systems of integrable P∆Es, including the lattice Boussinesq, Schwarzian Boussinesq, Toda-Modified Boussinesq systems, and the two-component potential Korteweg-de Vries system. Previously unknown Lax pairs are presented for systems of P∆Es derived by Hietarinta.

iii Lax pairs are not unique. To the contrary, for any P∆E there exists an inﬁnite number of Lax pairs due to gauge equivalence. The investigation of gauge and gauge-like transformations is a novel component of this thesis. A detailed discussion is given of how edge equations should be handled to obtain gauge and gauge-like equivalent Lax matrices of minimal size. The Lax pairs for Hietarinta’s systems presented in this thesis are compared with those computed by Zhang, Zhao, and Nijhoﬀ via a direct linearization method.

iv LIST OF ABBREVIATIONS

Adler,Bobenko,Surisclassiﬁcation ...... ABS

ConsistencyAroundtheCube...... CAC

DiscreteDiﬀerenceEquation...... DDE

Discrete/lattice Korteweg-de Vries Equation ...... lKdV

Discrete/lattice potential Korteweg-de Vries Equation ...... lpKdV

Discrete/lattice modiﬁed Korteweg-de Vries Equation ...... lmKdV

Discrete/lattice Schwarzian Korteweg-de Vries Equation ...... lsKdV

ContinuousKorteweg-deVriesEquation ...... KdV

Continuous potential Korteweg-de Vries Equation ...... pKdV

Continuous modiﬁed Korteweg-de Vries Equation ...... mKdV

PartialdiﬀerenceEquation...... P∆E

InverseScatteringTransform ...... IST

v ACKNOWLEDGMENTS

Many thanks to my advisor, Dr. Willy Hereman, for his guidance, his insight, and, perhaps most of all, his patience. I can confidently say that without his assistance, this dissertation would not have been possible. I also wish to thank the Department of Applied Mathematics and Statistics for their patience and support. Also, I would like to thank my friends and family for not asking me to explain my research and for not walking away when I tried. This research was supported in part by the National Science Foundation (NSF), under Grant No. CCF–0830783. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF.

vi For Robert and Sue.

vii TABLE OF CONTENTS

ABSTRACT ...... iii

LISTOFABBREVIATIONS ...... v

ACKNOWLEDGMENTS ...... vi

DEDICATION ...... vii

LISTOFFIGURES ...... xi

LISTOFTABLES ...... xii

CHAPTER1 INTRODUCTION ...... 1

CHAPTER2 LAXPAIRS ...... 6

2.1 LaxPairsofPDEs ...... 7

2.2 LaxPairsofP∆Es ...... 10

CHAPTER 3 ORIGINS OF PARTIAL DIFFERENCE EQUATIONS ...... 13

3.1 Discrete Completely Integrable Systems...... 16

3.2 B¨acklundTransformations ...... 16

3.2.1 Sine-Gordonequation...... 17

3.2.2 Korteweg-deVriesequation ...... 21

3.3 DirectLinearizationoftheKdVEquation ...... 26

3.3.1 DiscreteDirectLinearization...... 30

3.4 BilinearOperator ...... 33

3.4.1 DiscreteHirotaFormalism ...... 38

3.5 MultidimensionalConsistency ...... 42

viii 3.5.1 ConsistencyAroundtheCube ...... 42

3.5.2 ABSClassiﬁcation ...... 44

CHAPTER4 CONTINUUMLIMIT...... 50

4.1 Semi-DiscreteLimits ...... 50

4.2 StraightContinuumLimit ...... 51

CHAPTER 5 SYMBOLIC COMPUTATION OF LAX PAIRS ...... 55

5.1 ConsistencyaroundthecubeforscalarP∆Es ...... 55

5.2 ComputationofLaxpairsforscalarP∆Es ...... 57

5.3 Determination of the scalar factors for scalar P∆Es ...... 58

5.4 Consistency around the cube for systems of P∆Es ...... 60

5.5 ComputationofaLaxpairforsystemsofP∆Es ...... 63

5.6 Determination of the scalar factors for systems of P∆Es ...... 66

CHAPTER 6 GAUGE EQUIVALENCES ...... 72

6.1 Gauge Equivalence of Lax Pairs for PDEs and P∆Es ...... 72

6.2 DerivationofGaugeEquivalentLaxPairs ...... 73

6.3 GeneralizedHietarintaSystems ...... 80

6.3.1 A-2System ...... 80

6.3.2 B-2System ...... 85

6.3.3 C-3System ...... 86

6.3.4 C-4System ...... 89

CHAPTER7 SOFTWAREIMPLEMENTATION ...... 95

7.1 Algorithm ...... 95

7.1.1 ComputationofaLaxpair...... 96

ix 7.1.2 VeriﬁcationoftheLaxpair...... 97

7.2 SoftwarePackage ...... 98

7.2.1 LPSolve ...... 99

7.2.2 SampleLatticeFiles ...... 101

7.2.3 OutputDataFiles ...... 102

CHAPTER 8 CONCLUSIONS AND FUTURE DIRECTION ...... 105

8.1 FutureDirectionsandOpenQuestions ...... 105

REFERENCESCITED ...... 108

APPENDIX A LINEARIZING THE LPKDV EQUATION ...... 117

APPENDIXB LPKDVEQUATIONTOLKDVEQUATION ...... 118

APPENDIX C DIRECT LINEARIZATION OF KORTEWEG-DE VRIES EQUATION ...... 123

APPENDIX D HIROTA’S BILINEAR OPERATOR ...... 130

D.1 Hirota’s Bilinear Diﬀerential Operators ...... 130

D.2 Hirota’sDiﬀerenceOperator ...... 133

x LIST OF FIGURES

Figure 1.1 The P∆E is deﬁned on the simplest quadrilateral (a square)...... 1

Figure2.1 CommutingschemeforLaxequation ...... 10

Figure3.1 OriginsofP∆Es ...... 15

Figure 3.2 A visual representation of the Bianchi PermutabilityTheorem ...... 19

Figure3.3 TheP∆E isdeﬁnedonthecube...... 44

Figure3.4 Thetetrahedronproperty...... 46

Figure4.1 Straightlimit...... 52

Figure7.1 InitialDialogofLPSolve...... 100

Figure 7.2 Sample data ﬁle for the generalized C-3 equation ...... 102

Figure 7.3 Sample output of LPSolve for the Q1 equation ...... 103

Figure 7.4 Sample output of LPSolve for the Boussinesq system ofequations . . . . 104

FigureB.1 TheextendedlatticewithABSlabeling ...... 121

xi LIST OF TABLES

Table 2.1 Gauge equivalent Lax pairs for lattice Boussinesq system ...... 12

Table 3.1 The Q4 equation ...... 47

Table3.2 ScalarP∆EsofABSclassiﬁcation ...... 48

Table3.3 AdditionalScalarP∆Es ...... 49

Table5.1 EdgeConstrainedSystemsofP∆Es ...... 68

Table5.2 AdditionalSystemsofP∆Es ...... 70

Table6.1 GeneralizedHietarintaA-2System ...... 91

Table6.2 GeneralizedHietarintaB-2System ...... 92

Table6.3 GeneralizedHietarintaC-3System ...... 93

Table6.4 GeneralizedHietarintaC-4System ...... 94

xii CHAPTER 1 INTRODUCTION

This thesis is primarily devoted to the study and implementation of an algorithm to compute a Lax pair associated with a given nonlinear system of fully discrete integrable equations. These equations are also known in the literature as difference-difference equations or partial difference equations (P∆Es), a name we will most frequently use in this thesis. Specifically, for a given nonlinear P∆E or system thereof, we discuss the Consistency Around the Cube (CAC) property. This property, which is also called multi-dimensional consistency, determines the integrability of the equation(s). Using the algorithmic nature of the CAC test, we are able to compute Lax pairs in a straightforward way. During the construction process, we also investigate the variations and flexibility introduced by discrete systems with edge equations. Specifically, we will investigate P∆Es,

(u ,u ,u ,x ; p,q)=0, (1.1) F n,m n+1,m n,m+1 n+1,m+1 which are deﬁned on a 2-dimensional quad-graph as shown in Figure 1.1. The ﬁeld variable

x2 = un,m+1 x12 = un+1,m+1 p

q q

x = un,m x1 = un+1,m

Figure 1.1: The P∆E is deﬁned on the simplest quadrilateral (a square).

x = un,m depends on lattice variables n and m. A shift of x in the horizontal direction (the 1 direction) is denoted by x u , a shift in the vertical or 2 direction by x u − 1 ≡ n+1,m − 2 ≡ n,m+1

1 and a shift in both directions by x u . Furthermore, depends on the lattice 12 ≡ n+1,m+1 F parameters p and q which correspond to the edges of the quadrilateral. In the simpliﬁed notation (1.1) is replaced by

(x,x ,x ,x ; p,q)=0. (1.2) F 1 2 12

Alternate notations are used in the literature. For instance, many authors denote (x,x1,x2,x12) by (x, x,˜ x,ˆ x˜ˆ) while others use (x00,x10,x01,x11). As an example of the type of equations we will address, consider the lattice potential Korteweg-de Vries (lpKdV) equation, in various notations we just mentioned,

(p q + u u )(p + q u + u )= p2 q2, (1.3a) − n,m+1 − n+1,m − n+1,m+1 n,m − (p q + u u )(p + q u + u )= p2 q2, (1.3b) − 01 − 10 − 11 00 − or simply,

(p q + x x )(p + q x + x)= p2 q2. (1.3c) − 2 − 1 − 12 −

The advantage of the first two notations is that it allows one to consider, if necessary, nodes on a lattice extended beyond that depicted in Figure 1.1 such as un 1,m or un,m+2. The notation − in (1.3c) is somewhat simpler and sufficient for the primary focus of this dissertation – those P∆Es that are defined on the square and possess the CAC property. Prior to an in-depth discussion of the Lax pair algorithm and its implementation, it is appropriate to first understand the fundamentals in the context of partial differential equations (PDEs). In particular, a brief historical discussion of Lax pairs for both PDEs and P∆Es is given in Chapter 2. The historical context provides an additional understanding of the connections between PDEs and P∆Es and the importance of Lax pairs. A second aim of this dissertation is a thorough investigation and consolidation of the research which led to the discovery and development of integrable P∆Es. As discussed, Lax pairs for semi-discrete systems and doubly-discrete systems (P∆Es) first appeared in work by Ablowitz and Ladik [1–4] who were concerned with discretizing PDEs, both in time

2 and space, without destroying their integrability properties. However, P∆Es also appeared early on in papers by Hirota [5–9] covering a soliton preserving discretization of his bilinear method for nonlinear PDEs. In addition, Miura [10] and Wahlquist & Estabrook [11] indi- rectly contributed to the development of the theory of P∆Es through their work on Bäcklund transformations. A major contribution to the study of P∆Es came from Nijhoff and col- leagues [12–14]. Under the supervision of Capel, the Dutch research group used a direct linearization method and Bäcklund transformations, in connection with a discretization of the plane wave factor, to derive several P∆Es. These methods as well as the geometry-based classification by Adler, Bobenko and Suris [15, 16] are discussed in detail in Chapter 3. A significant portion of the research for this dissertation was devoted to the development of the software package LaxPairPartialDifferenceEquations.m [17, 18]. The package is written in Mathematica syntax. It does the symbolic computations to get Lax pairs automatically, i.e., without intervention of the user. The software extends the functionality of the code developed by [19] for scalar P∆Es to systems of P∆Es. It contains the algorithms to test the CAC property and compute Lax pairs for systems of P∆Es defined on the square, including systems with edge equations. LaxPairPartialDifferenceEquations.m [18] is a stand-alone package that includes an extensive user interface allowing a user to verify the consistency around the CAC property, and either compute a Lax pair or test the validity of a user-provided Lax pair by substitution into the Lax equation. Our package was used to compute Lax pairs for an extensive collection of P∆Es which are known to be consistent around the cube. The Lax pairs that were already available in the literature provided a data base of test cases during the development of the software. In turn, using our new code we were able to compute Lax pairs for families of P∆Es of recent origin, in particular, systems of P∆Es recently discovered by Hietarinta [20]. The new results obtained with our software are presented in several tables in this dissertation. For the interested reader, the information presented in this dissertation is also intended to extend, complement or support the research from several theses. The following non-

3 comprehensive list of theses provided the author with additional insights and information incorporated into this dissertation. Additional discussions regarding integrability of P∆Es are provided in [21–23]. Further discussions of Bäcklund and Darboux transformations as well as the Inverse Scattering Transform are provided in [24–26]. The Lagrangian structure of P∆Es is discussed in [27, 28]. Additional information on the direct linearization method is provided in [29], The outline of this thesis is as follows: In Chapter 2 we provide a brief discussion of Lax pairs for both PDEs and P∆Es. Chapter 3 gives an overview of some of the methods for deriving integrable discretizations of completely integrable nonlinear PDEs. Specifically, we will use PDEs of KdV-type as a basis to derive the corresponding P∆Es using Bäcklund transformations [11], Hirota’s bilinear method [5], and the direct linearization approach by Nijhoff et al. [13] and Quispel et al. [30]. We also briefly discussion the seminal work of Adler et al. [16]. Using discrete differential geometry (see [31]), their search for integrable P∆Es resulted in the famous Adler, Bobenko & Suris (ABS) classification of discrete integrable systems. In Chapter 4 we discuss the recovery of the underlying PDE from the matching P∆E by use of a continuum limit. At that point our discussion of one particular P∆E, namely the lpKdV equation, has gone full circle – from the continuous PDE to a fully discrete P∆E via the methods discussed in Chapter 3 and then back to the continuous PDE via a continuum limit. That example illustrates the intimate connections between continuous and discrete integrable systems. Chapter 5 covers the findings in [32] where we presented the algorithms for determining if a given system of P∆Es is consistent around the cube and the actual computation of Lax pairs. Chapter 5 also includes a discussion of how to handle edge equations and their impact on the resulting Lax pair. Chapter 6 is based on [33] which includes a detailed discussion of gauge and gauge-like equivalences that were discovered during our research on generalized Hietarinta systems. A brief overview of the functionality, scope, and limitations

4 of the software package LaxPairPartialDifferenceEquations.m is given in Chapter 7. The package was used to generate Lax pairs for a significant number of P∆Es, including those explicitly covered in this dissertation as well as other P∆Es encountered during the research. These results are presented in the various tables throughout this dissertation. In Chapter 8 we summarize the findings of the thesis and discuss future areas of research, including possible enhancements and extensions of the software package. Finally, we clarify the original contributions made by the author. They include (i) the software package LaxPairPartialDifferenceEquations.m which supports verification of the CAC property, computation of Lax pairs and validation of existing Lax pairs for both scalar and systems of P∆Es, as discussed in Chapter 7. The author is sole developer of the software. (ii) The research discussed in Chapter 5 is based on a published collaboration [32] between the author, Hereman, Quispel, and van der Kamp, to which the author was the primary contributor. (iii) The work discussed in Chapter 6 is based on an unpublished collaboration [33] with Hereman to which the author is also the primary contributor.

5 CHAPTER 2 LAX PAIRS

The complete integrability of a system of PDEs has been an area of active research with many approaches advocated over the years. One concept that has consistently appeared in several of the various approaches is the Lax pair – a reformulation of the given nonlinear equations as a compatibility condition for a system of linear equations [34]. The very first Lax pair [35] is a duo of commuting linear differential operators representing the Korteweg-de Vries (KdV) equation. Lax’s idea was to replace a nonlinear PDE, such as the KdV equation, by a pair of linear PDEs of high-order (in an auxiliary eigenfunction) whose compatibility requires that the nonlinear PDE holds. One can write these high-order linear PDEs as a system of PDEs of first order; hence, replacing the Lax operators with a pair of matrices. The Lax equation to be satisfied by these matrices is commonly referred to as the zero- curvature representation [36] of the nonlinear PDE. The discovery of Lax pairs was crucial for the further development of the inverse scattering transform (IST) method which had been introduced in [37]. Similarly, in the shift from continuous to discrete integrable systems, the concept of integrability proved to also be an active area of research. For P∆Es, Lax pairs first appeared in the work of Ablowitz and Ladik [2, 4], and subsequently in [13] for other equations. The fundamental characterization of integrable P∆Es as being multi-dimensionally consistent [15, 38] is intimately related to the existence of a Lax pair. Lax pairs for P∆Es are not only crucial for applying the IST, they can be used to construct integrals for mappings and correspondences obtained as periodic reductions, using the so-called staircase method. This method was developed in [39] and extended in [40] to cover more general reductions. Essential to the staircase method is the construction of a product of Lax matrices (the monodromy matrix) whose characteristic polynomial is an

6 invariant of the evolution. In fact, the monodromy matrix can be interpreted as one of the Lax matrices for the reduced mapping [41–43]. Through expansion of the characteristic equation of the monodromy matrix in the spectral parameters a number of functionally independent invariants can be obtained. A recent investigation [44] supports the idea that the staircase method provides suﬃciently many integrals for the periodic reductions to be completely integrable (in the sense of Liouville-Arnold).

2.1 Lax Pairs of PDEs

Lax showed [35] that a completely integrable nonlinear PDE can be associated with a system of linear PDEs in an auxiliary function ψ(x,t),

ψ = λψ, (2.1) L ψ = ψ, t M where and are linear diﬀerential operators. The operators ( , ) are known as a Lax L M L M pair for the given PDE. Consider, for example, the ubiquitous KdV equation [45] for u(x,t),

ut + αuux + uxxx =0, (2.2) where α is any non-zero real constant. A Lax pair for (2.2) is given by [35]

1 = D2 + αu I, (2.3) L x 6 1 = 4D3 αu D αu I M − x − x − 2 x

D Dn where x is the total derivative operator with respect to x [34], and x denotes repeated application of Dx (n times) and I is the identity operator. As discussed in [34], substitution of and into (2.1) yields L M

7 1 D2ψ = λ αu ψ, (2.4) x − 6 1 D ψ = 4D3ψ αu D ψ αu ψ, (2.5) t − x − x − 2 x

with corresponding compatibility condition of 1 D D2ψ D2D ψ = α (u + αuu + u ) ψ =0, (2.6) t x − x t 6 t x 3x when (2.2) is taken into account. In general [34],

+[ , ] ˙= O, (2.7) Lt L M where ˙=denotes that the equation holds for solutions of the given nonlinear PDE. Here [ , ] is the commutator of the operators and O is the zero operator. L M ≡ LM−ML Lax’s technique was further extended, ﬁrst in 1972 by Zakharov and Shabat [46], and then by Ablowitz et al. [47] with the introduction of a matrix formalism for Lax pairs. They associated matrices X and T with the operators and forming a compatible linear L M system,

Φx = XΦ and Φt = TΦ, (2.8) with vector function Φ(x,t). One can readily show [34] that the compatibility condition for (2.8) is the (matrix) Lax equation (also known as the zero curvature condition),

X T +[X, T] ˙= 0. (2.9) t − x Here, [X, T] := XT TX is the matrix commutator. − Returning to the KdV equation, it is well known (see, e.g., [34]) that

0 1 X =   (2.10a) λ 1 αu 0  − 6   

8 and

1 αu 4λ 1 αu 6 x − − 3 T =   (2.10b) 4λ2 + 1 αλu + 1 α2 u2 + 1 αu 1 αu  − 3 18 6 xx − 6 x    form a Lax pair for (2.2). In this example

ψ Φ= , (2.11) ψx! where ψ(x,t) is the scalar eigenfunction of the Schr¨odinger equation [48],

ψ (λ 1 αu)ψ =0, (2.12) xx − − 6 with eigenvalue λ and potential proportional to u(x,t). It has been shown (see, e.g., [34] or [49, p. 22]) that if (X, T) is a Lax pair, then so is (Xˆ , Tˆ ) where

1 1 1 1 Xˆ = GXG− + GxG− and Tˆ = GTG− + GtG− , (2.13)

for an arbitrary invertible matrix G of the correct size. The above transformation comes from changing Φ in (2.8) into Φ=ˆ GΦ and requiring that

Φˆ x = Xˆ Φˆ and Φˆ t = Tˆ Φˆ. (2.14)

In physics, transformations like (2.13) are called gauge transformations. Obviously, a Lax pair for a given PDE is not unique. In fact, there exists an inﬁnite number of Lax pairs which are gauge equivalent through (2.13). In the case of the KdV equation, for example using the gauge matrix

ik 1 − G =   , (2.15) 1 0  −    we see that (2.10) is gauge equivalent to the Lax pair,

1 ik 6 αu ˆ − X =   (2.16a) 1 ik  −   

9 and

4ik3 + 1 iα k u 1 αu 1 α 2k2 u 1 αu2 + ik u 1 u − 3 − 6 x 3 − 6 x − 2 xx Tˆ =   , (2.16b) 4k2 + 1 αu 4ik3 1 iα k u + 1 αu  − 3 − 3 6 x    where λ = k2. The latter Lax matrices are complex matrices. However, in (2.16a) the − eigenvalue k appears in the diagonal entries which is advantageous if one applies the Inverse Scattering Transform to solve the initial value problem for the KdV equation.

2.2 Lax Pairs of P∆Es

In analogy with the deﬁnition of Lax pairs (in matrix form) for PDEs, a Lax pair for a P∆E is a pair of matrices, (L,M), such that the compatibility of the linear equations, for an auxiliary vector function ψ,

ψ1 = Lψ, (2.17a)

ψ2 = Mψ, (2.17b)

is equivalent to the P∆E. Recall that ψ stands for ψn,m and ψ1 denotes ψn+1,m. Likewise, ψ2

denotes ψn,m+1. The crux is to ﬁnd suitable matrices L and M so that the nonlinear P∆E can be replaced by (2.17a)-(2.17b). To avoid trivial cases, the compatibility of (2.17a) and (2.17b) should only hold on solutions of the given nonlinear P∆E.

L2 ψ2 ψ12

M M1

L ψ ψ1

Figure 2.1: Commuting scheme for Lax equation

10 The compatibility of (2.17a) and (2.17b) can be readily expressed as follows:

Shift (2.17a) in the 2 direction, i.e., ψ = L ψ = L Mψ; • − 12 2 2 2 Shift (2.17b) in the 1 direction, i.e., ψ = ψ = M ψ = M Lψ; and • − 21 12 1 1 1 Equate the results. •

1 Hence, L2Mψ = M1Lψ must hold on solutions of the P∆E. The compatibility is visualized in Figure 2.1, where commutation of the scheme indeed requires that L2M = M1L. The corresponding Lax equation is thus

L M M L ˙=0, (2.18) 2 − 1 where ˙=denotes that the equation holds for solutions of the P∆E. As with completely integrable PDEs, Lax pairs of P∆Es are not unique for they are equivalent under gauge transformations. As given in Table 2.1, if (L,M) is a Lax pair then so is ( , ) where L M 1 1 = L − , = M − , (2.19) L G1 G M G2 G for any arbitrary non-singular matrix . Indeed, ( , ) satisfy ˙= 0, which G L M L2M−M1L 1 follows from (2.18) by pre-multiplication by and post-multiplication by − . Alterna- G12 G tively, φ = φ and φ = φ, provided φ = ψ. The Lax pairs (L,M) and ( , ) are said 1 L 2 M G L M to be gauge equivalent. Table 2.1 gives an example of gauge equivalent Lax pairs. Alternative interpretations of the edge equations given in the lattice Boussinesq system of equations (see Table 5.1) yield diﬀering Lax pairs. The corresponding gauge matrix relating the diﬀerent Lax pairs is also given. Further examples of gauge equivalent Lax pairs are discussed in detail in Chapter 6.

1 In the diagram M1 denotes the shift of M in the 1 direction (horizontally) while L2 denotes the shift of L in the 2 direction (vertically). − −

11 Table 2.1: Gauge equivalent Lax pairs for lattice Boussinesq system

Lattice z xx + y = 0, z xx + y = 0, 1 − 1 2 − 2 Boussinesq (x x )(z xx + y ) p + q = 0. 2 − 1 − 12 12 −

y xx1 0 1 y xx2 0 1 1 − 1 − F L= ℓ x2 z , M= ℓ x2 z , x  21 −  x  21 −  ψLM = g yy xx y yy xx y    − 1 1 − 1  − 2 2 − 2 h       where where   ℓ = z(xx y) x2y + x(p k). ℓ = z(xx y) x2y + x(q k). 21 1 − − 1 − 21 2 − − 2 −

x1 1 0 x2 1 0 F − − L M =  y1 0 1 , =  y2 0 1 , ψLM = f  − − ℓ31 z x ℓ31 z x g  −   −        where ℓ = zx xy + p k. where ℓ = zx xy + q k.   31 1 − 1 − 31 2 − 2 −

1 0 0 1 0 0 G y 1 , H G 1 , =  x 0 x  = − =  0 0 1 0 1 0 y x 0   −    1  1  where L = G1LG− . where L = H1LH− .

12 CHAPTER 3 ORIGINS OF PARTIAL DIFFERENCE EQUATIONS

As we are concerned with discrete integrable systems, it is logical that our investigation into the origin stems from work done on nonlinear waves and continuous integrable systems, speciﬁcally, the Korteweg-de Vries (KdV) equation. As originally published in 1895 in the Philosophical Magazine [50] to describe shallow water waves, an equation for u(x,t) of the form

ut + αuux + βuxxx =0, (3.1) for some constants2 α and β, is called the Korteweg-de Vries equation3. Speciﬁcally, for the discussions to follow, we will consider,

ut +6uux + uxxx =0. (3.2)

Within the family of KdV equations we will address the potential Korteweg-de Vries (pKdV) equation which may be written as,

w 3w2 + w =0, (3.3) t − x xxx by substituting u = w into (3.2) and integrating with respect to x or as, − x 2 wt +3wx + wxxx =0, (3.4) from the substitution u = w . Also, substituting u = v v2 into (3.2), as shown in Section x x − 3.2.2, yields the modiﬁed Korteweg-de Vries (mKdV) equation,

v 6v2v + v =0. (3.5) t − x xxx

2These coeﬃcients are arbitrary in that they can be scaled into any numerical values, such as α = 6 and β = 1. ∂u ∂u 3In (3.1) and all other PDEs in this thesis, the subscripts denote partial derivatives, e.g. u = , u = t ∂t x ∂x ∂3u and u = xxx ∂x3

13 The KdV equation experienced a renewed interest with the publication by Gardner, Greene, Kruskal and Miura [37] which introduced a new method for finding solutions to (3.2). This method, the Inverse Scattering Transform, is the nonlinear analogue of the Fourier transform and gave impetus for more general schemes applicable to other, exactly integrable, nonlinear PDEs, such as the nonlinear Schrödinger and sine-Gordon equations. Along with this method came the discovery of solitary wave and soliton solutions and the advent of a long and ongoing study of integrable systems. Subsequently, based on the work done by Gardner et al., Lax proposed a more general framework for finding soliton solutions to nonlinear evolution equations [35], i.e., equations of the form,

ut = F (u), u = u(x,t), (3.6) where F is a nonlinear diﬀerential operator which is independent of ∂t. His technique involved relating the original nonlinear PDE to two linear operators, hereafter referred to as a Lax pair, via a compatibility condition. As discussed in Chapter 2, the technique was further extended [46, 47], with the introduction of the matrix representation of the Lax pair,

ψx = Lψ, (3.7)

ψt = Mψ, and corresponding compatibility condition,

L M +[L,M] ˙=0, (3.8) t − x referred to as the Lax equation. The Lax pair and the corresponding Lax equation became key tools to study nonlinear integrable PDEs. However, in as much as the focus of this dissertation is discrete integrable equations, more recently there has also been a shift in focus from continuous to discrete integrable systems as well. For example, in 1995, Kruskal stated (cited in [51]) “For years we have been thinking

14 that the integrable evolution equations are fundamental ones. It is becoming clear now that the fundamental objects are integrable discrete equations”. Thus, much research has been done on discretizing continuous integrable systems in such a way as to preserve integrability in the resulting diﬀerences equations.

Num. Discretization B¨acklund Transforms

B¨acklund Transforms PDEs P∆Es

Direct Discrete

Linearization Plane Wave

Bilinear Operator ABS Classiﬁcation

Figure 3.1: Origins of P∆Es

In this chapter, we discuss several methods, as shown in Figure 3.1, used to derive the P∆Es that are the focus of this dissertation. These methods stem from:

Hirota’s work from the 1970s [5–9] which focused on discretizing bilinear equations; • work of Wahlquist and Estabrook [11] (also from the early 1970s) and their discovery • of a Bäcklund transformation for the pKdV equation which established a connection between these Bäcklund transformations and Hirota’s multi-soliton solutions; research by Ablowitz, Kaup, Newell and Segur [52] of the same era, who applied the • Inverse Scattering Transform first to solve the sine-Gordon equation. They showed [47] that equations such as the sine-Gordon and nonlinear Schrödinger equations fit into a single matrix eigenvalue problem. When discretized in space and time, that eigenvalue problem gives rise to integrable partial difference equations [3, 4]; work from the 1980s by Capel, Nijhoff, Quispel and van der Linden [12, 13, 53] who, in • building on the findings by Ablowitz et al., established Bäcklund transformations of so-

15 lutions to the direct linearization equation, and as a separate investigation, discretized the plane wave factor involved in the direct linearization equation.

In addition to these, we also discuss the geometric approach by Adler, Bobenko and Suris [16] who established Consistency Around the Cube (CAC), discovered independently by Nijhoﬀ and Walker [54], as a property to identify integrable P∆Es. Though not necessarily an exhaustive list of the sources for the P∆Es considered in this dissertation, the research addressed above does provide historical insight into the research which accompanied the growing interest in discrete integrable equations.

3.1 Discrete Completely Integrable Systems

As shown in [3, 4], a discrete analogue of the Zakharov-Shabat eigenvalue problem allows one to isolate nonlinear P∆Es which are still solvable with the Inverse Scattering Transform method (see also [48, p. 114], [55] and [56]). Ablowitz and Ladik were primarily concerned with discretizing PDEs, first in space alone [1, 2] and later in both time and space [3, 4], without destroying their integrability properties, i.e., having an associate linear operator (Lax) pair, preserving an infinite set of (discrete) conservation laws and soliton solutions. The possible application to the numerical simulation of solutions of the original PDE was of secondary concern. The papers by Ablowitz and Ladik dealt with the nonlinear Schrödinger equation explicitly but the same method can be applied to the KdV and mKdV equations, leading to fully discrete versions of these equations. For an in-depth discussion of these nonlinear P∆Es, mainly from the perspective of numerical analysis, we refer to papers by Ablowitz and Taha [57–59]. The P∆Es appearing in the work by Ablowitz, Ladik, and Taha are not consistent around the cube and therefore outside the scope of this Dissertation.

3.2 B¨acklund Transformations

In 1880 Bäcklund introduced a transformation of pseudospherical4 surfaces [60]. Though originally introduced in the context of differential geometry, the Bäcklundtransformation

4A surface with constant negative Gaussian curvature.

16 assumed a more prominent role in the discussion of discrete equations with the application of the Bianchi permutability property [61]. In short, for the purpose of the following discussion, a Bäcklund transformation relates solutions of different PDEs. Similarly, an auto-Bäcklund transformation relates solutions of the same PDE.

3.2.1 Sine-Gordon equation

Consider the PDE ﬁrst used by B¨acklund himself, namely, the sine-Gordon equation,

ϕ ϕ = sin ϕ, (3.9) XX − TT in space-time coordinates where ϕ = ϕ(X,T ). Converting (3.9) into light cone coordinates5 gives the, perhaps more familiar form,

uxt = sin u, (3.10) where u(x,t) is the angle between asymptotic lines of the pseudospherical surface. Now consider the system of equations,

u + v (u v) =2α sin , (3.11a) − x 2 2 u v (u + v) = sin − , (3.11b) t α 2 where α = 0 is an arbitrary parameter. Cross diﬀerentiation then gives 6

u + v u v u + v (u v) = α(u + v) cos = 2sin − cos , and (3.12) − xt t 2 2 2 1 u v u + v u v (u + v) = (u v) cos − = 2sin cos − . (3.13) xt α − x 2 2 2 Adding (3.12) and (3.13) gives

u v u + v u + v u v u = sin − cos + sin cos − , (3.14) xt 2 2 2 2

5In special relativity, light cone coordinates [62] is a special coordinate system where two of the coordinates are null coordinates and all the other coordinates are spatial.

17 and subtracting (3.12) from (3.13) gives

u + v u v u v u + v v = sin cos − sin − cos . (3.15) xt 2 2 − 2 2 Reduction of (3.14) and (3.15) via trigonometric identities yields,

uxt = sin u and vxt = sin v. (3.16)

As both u and v must then also satisfy the sine-Gordon equation, (3.11) constitutes an auto- B¨acklund transformation for the equation. This system can then be used to generate a new solution from a given a solution for (3.10). For example, consider a trivial solution to (3.10), u(x,t) = 0. Substitution into (3.11a) gives v = 2α sin v , which, in turn, yields x − 2 dv v 2αx = = 2ln tan + f(t) (3.17) − sin v − 4 Z 2 2 v for an arbitrary function f(t). Similarly, substitution into (3.11b) gives vt = sin − α 2 which yields 2t dv v = = 2ln tan + g(x), (3.18) α − sin v − 4 Z 2

for an arbitrary function g(x). Subtracting (3.17) from (3.18) 2t 2t 2αx = g(x) f(t), or rather, g(x) = f(t) 2αx. (3.19) α − − − α − Then f(t) and g(x) may be deﬁned as 2t f(t)= + K and g(x)= 2αx + K, (3.20) − α − for some arbitrary constant K. Thus, both (3.17) and (3.18) become v 2t 2ln tan = 2αx + K. (3.21) 4 − α −

Solving,

(t/α+αx) v(x,t) = 4arctan Ce− , (3.22) for C = eK/2. It is easily veriﬁed that v(x,t) is also a solution for (3.10). Thus, even though (3.11) cannot be solved explicitly, the system can be used to generate new solutions from,

18 potentially, trivial solutions. This process can be repeated beginning with (3.22) to generate yet another solution.

u(1) α β

u u(21) = u(12)

β α u(2)

Figure 3.2: A visual representation of the Bianchi Permutability Theorem

In general, for an arbitrary solution u of (3.10), an auto-B¨acklund transformation with parameter α can be used to generate a solution, u(1). Similarly, from u(1), the transformation with parameter β, can be used to generate u(21). Repeating this process with the parameters in reverse order, starting again from u, generates solutions u(2) and then u(12). Bianchi’s Theorem of Permutability [63] states that u(12) = u(12), which is diagrammatically shown in Figure 3.2. This consistency allows for the construction of a discrete lattice equation corresponding to (3.10). Consider, as outlined in [64], (3.11) with parameter α, is given as

u + u(1) 2 u u(1) (u u(1)) =2α sin and (u + u(1)) = sin − . (3.23) − x 2 t α 2 Similarly, with the transformation parameter β, (3.11) is given as

u + u(2) 2 u u(2) (u u(2)) =2β sin and (u + u(2)) = sin − . (3.24) − x 2 t β 2 Then, applying the transformation (3.24) to the solution, u(1), generated by (3.23) gives

u(1) + u(21) 2 u(1) u(21) (u(1) u(21)) =2β sin and (u(1) + u(21)) = sin − . (3.25) − x 2 t β 2

19 Applying the transformation (3.23) to the solution, u(2), generated by (3.24) gives

u(2) + u(12) 2 u(2) u(12) (u(2) u(12)) =2α sin and (u(2) + u(12)) = sin − . (3.26) − x 2 t α 2 (1) (1) References to ux and ut may then be removed by substitution of (3.23) into (3.25) giving

u + u(1) u(1) + u(21) u(21) = u 2α sin 2β sin , (3.27a) x x − 2 − 2 2 u u(1) 2 u(1) u(21) u(21) = u sin − + sin − . (3.27b) t t − α 2 β 2 (2) (2) Similarly, substitution of (3.24) into (3.26) removes ux and ut yielding

u(2) + u(12) u + u(2) u(12) = u 2α sin 2β sin , (3.28a) x x − 2 − 2 2 u(2) u(12) 2 u u(2) u(12) = u + sin − sin − . (3.28b) t t α 2 − β 2 Application of the permutability property, i.e., u(12) = u(21), and some trigonometric identities, then yield

u + u(1) u(2) u(12) u u(1) + u(2) u(12) α sin − − = β sin − − . (3.29) 4 4 Deﬁning,

i i i i x := exp u , x := exp u(1) , x := exp u(2) , and x := exp u(12) , (3.30) 2 1 2 2 2 12 2 (3.29) becomes

1 1 1 1 1 1 1 1 α √x√x1 √x2√x12 = β √x √x2 √x1 √x12 √x √x − √x √x √x √x − √x √x 2 12 1 1 12 2 (3.31) which simpliﬁes into

α(xx x x )= β(xx x x ). (3.32) 1 − 2 12 2 − 1 12

20 As we shall see in subsequent sections, (3.32) is an example of a P∆E– the primary focus of this dissertation. In particular, (3.32) is the discrete sine-Gordon equation.

3.2.2 Korteweg-de Vries equation

Returning to the KdV equation (3.2), the Bäcklund transformation can be used to derive a solitary wave solution for this equation, and, as with the sine-Gordon equation above, it may be used to derive an equivalent discrete difference equation. However, before discussing the results from the corresponding Bäcklund transformation, it is important to first discuss a transformation specific to the KdV equation which leads to the Bäcklund equation and provides some insight into solutions of the KdV equation. In 1968 Miura [10] discovered a “remarkable transformation” that relates solutions of the KdV equation and the so-called modified Korteweg-de Vries equation, i.e.,

v 6v2v + v =0. (3.33) t − x xxx Consider the function

u = v v2. (3.34) x − Substitution into the KdV equation (3.2) gives,

(v 2vv )+6(v v2)(v 2vv )+(v 6v v 2vv ) (3.35) xt − t x − xx − x 4x − x xx − xxx = 2v(v 6v2v + v )+(v 6v2v 12vv2 + v ) (3.36) − t − x xxx xt − 2x − x 4x ∂ =( 2v + )(v 6v2v + v )=0. (3.37) − ∂x t − x xxx

Thus, if v is a solution to the mKdV equation, then u, defined per the Miura transformation, (3.34), is a solution to the KdV equation. Additionally, given the relationship established between solutions of the KdV and mKdV equations, (3.34) together with (3.33) may be viewed as a Bäcklund transformation. Then in 1973, in efforts to generalize the approach to solving evolution equations such as the Burgers, sine-Gordon and KdV equations, Wahlquist and Estabrook [11] introduced a more elegant Bäcklund transformation based on the transformation (3.34) and the observa-

21 tion that (3.2) is Galilean invariant. For example, application of the Galilean transformation,

x˜ = x 6kt, t˜= t, andu ˜(˜x, t˜)= u(x,t) k, k R, (3.38) − − ∈ to (3.2) yields

u˜t˜ +6˜uu˜x˜ +˜ux˜x˜x˜ =0. (3.39)

This is readily veriﬁed since

u =u ˜˜ 6ku˜ , u =u ˜ , u =u ˜ , and 6uu = 6(˜u + k)˜u , (3.40) t t − x˜ x x˜ xxx x˜x˜x˜ x x˜ obtained by repeated applications of the chain rule. Thusu ˜(˜x, t˜) is also a solution to (3.2) and, as such, this property is called Galilean invariance. Now consider the Miura transformation, modiﬁed from (3.34) using the Galilean transformation (3.38) with k = p2,

u p2 = v v2 or rather u = v v2 + p2, (3.41) − x − x − for some real parameter p. Substitution into (3.2) gives ∂ ( 2v + )(v 6v2v + v +6p2v )=0, (3.42) − ∂x t − x xxx x which is a slightly modiﬁed version of (3.37). Thus, (3.41) maps solutions of

v 6v2v + v +6p2v =0 (3.43) t − x xxx x to solutions of the KdV equation6. Note that (3.43) is invariant under v v. Let 7→ − u := v v2 + p2 and u(1) := v v2 + p2 (3.44) − x − x − deﬁne two solutions, u and u(1), to (3.2) corresponding to solutions v and v, respectively, − of (3.43). Adding and subtracting these solutions gives,

u + u(1) = 2v2 +2p2 and u u(1) = 2v . (3.45) − − − x Now introduce the potentials w and w(1) by setting

(1) (1) u = wx and u = wx . (3.46)

6Note that (3.43) is the mKdV equation if p = 0.

22 Then w(x,t) and w(1)(x,t) are solutions to the potential Korteweg-de Vries equation,

2 wt +3wx + wxxx =0, (3.47) ignoring the integration constant. In terms of w, (3.45) becomes

(w + w(1)) = 2v2 +2p2, and w w(1) = 2v, (3.48) x − − − again ignoring the integration constant. Eliminating v yields

2(w + w(1)) =4p2 (w w(1))2. (3.49) x − − As both w and w(1) are solutions to the pKdV equation, (3.49) is then an auto-B¨acklund transformation for the pKdV equation (3.47). Thus, given a solution w, (3.49) may be used to compute a new solution w(1). For example, w(x,t) = 0 is clearly a (trivial) solution for (3.47). Then (3.49) becomes

2w(1) =4p2 (w(1))2, (3.50) x − v a non-linear Riccati equation. Using the substitution of w(1) =2 x , (3.50) becomes v 2 vxx = p v, (3.51)

px px which can be readily solved, yielding v = c1(t)e + c2(t)e− . Thus, the solution to (3.50) is

px px c (t)e c (t)e− w(1) =2p 1 − 2 , (3.52) c (t)epx + c (t)e px 1 2 − c(t)e2px 1 =2p − , where c(t)= c (t)/c (t), (3.53) c(t)e2px +1 1 2 e2px+κ(t) 1 =2p − , where κ(t)=ln c(t) , (3.54) e2px+κ(t) +1 1 =2p tanh(px + k(t)) , where k(t)= κ(t ). (3.55) 2 (1) (1) (1) (1) Then, upon substitution of the derivatives, wt , wx and wxxx, (3.47) for w reduces to

2 3 2p sech (px + k(t)) 4p + k′(t) =0. (3.56)

23 Then, k(t) = 4p3t + x , for some constant x . Thus, the solution for the pKdV equation − 0 0 derived using the B¨acklund transformation is

w(1)(x,t)=2p tanh px 4p3t + x . (3.57) − 0 Using (3.46) it follows that a solution for the KdV equation is

u(1)(x,t)=2p2sech2 px 4p3t + x . (3.58) − 0 As seen with the Bäcklund transformation for the sine-Gordon equation, this process can also be repeated to generate additional solutions for the pKdV and KdV equations. Returning to the auto-Bäcklund transformation (3.49) for the pKdV equation, which also conforms to the Bianchi permutability, a discrete lattice equation may be constructed by duplicating the process described previously for the sine-Gordon equation. For an arbitrary solution u(x,t) of (3.47), the auto-Bäcklund transformation (3.49) with parameter α can be used to generate a solution, u(1), and with parameter β to generate a solution u(2). The solutions u(1) and u(2) are connected by

2(u + u(1)) = α (u u(1))2, (3.59a) x − − and

2(u + u(2)) = β (u u(2))2. (3.59b) x − −

Repeated use of these auto-B¨acklund transformations on u(1) and u(2) yields

2(u(1) + u(21)) = β (u(1) u(21))2, (3.60a) x − − and

2(u(2) + u(12)) = α (u(2) u(12))2. (3.60b) x − −

Under the assumption of permutability, the diﬀerence of (3.60a) and (3.60b) yields

2(u(1) u(2)) = β α (u(1) u(12))2 +(u(2) u(12))2 (3.61) − x − − − −

24 which implies

2 2 2(u(1) u(2)) = β α +2u(12)(u(1) u(2))+(u(2) u(1) ). (3.62) − x − − − Similarly, the diﬀerence of (3.59a) and (3.59b) yields

2 2 2(u(1) u(2)) = α β +2u(u(1) u(2))+(u(2) u(1) ). (3.63) − x − − − Combining (3.62) and (3.63) then gives

(u(12) u)(u(1) u(2))= α β. (3.64) − − − Given the source PDE used for the derivation, (3.64) is known as the discrete or lattice pKdV equation and, is more commonly written, using the simpliﬁed notation, as

(x x)(x x )= p2 q2, (3.65) 12 − 1 − 2 − with α = p2 and β = q2. Interestingly, as shown in Appendix B, with the substitution v = x x , (3.65) becomes, 1 − 2

2 2 1 1 (v v12)=(p q )( ), (3.66) − − v2 − v1 the lattice KdV equation. As with the discrete sine-Gordon equation (3.32), both the discrete pKdV equation (3.65) and the lattice KdV equation (3.66) are examples of P∆Es which will be discussed in later

chapters. Speciﬁcally, (3.65) can be seen in Table 3.2, referred to as the H1 equation. However, unlike the lattice sine-Gordon equation and the lattice KdV equation, the discrete pKdV equation is an example of a P∆E of critical importance to this dissertation in that it will be shown to be integrable, as deﬁned in Section 3.5, which forms the primary focus of investigation and implementation of the software which accompanies this dissertation.

25 3.3 Direct Linearization of the KdV Equation

Fokas and Ablowitz [65, 66] have shown that if ϕ(x,t; k) solves the singular linear integral equation,

ϕ(x,t; ℓ) ϕ(x,t; k)+ iρ(x,t; k) dλ(ℓ)= ρ(x,t; k) (3.67) ℓ + k ZC with plane-wave factor

3 ρ(x,t; k)=ei(kx+k t) (3.68) and where dλ(k) and C are an appropriate measure and contour, respectively, then

w(x,t)= ϕ(x,t; k)dλ(k) (3.69) ZC satisﬁes the potential KdV equation,

w 3w2 + w =0, (3.70) t − x 3x and therefore

u = w = ∂ ϕ(x,t; k)dλ(k) (3.71) − x − x ZC satisﬁes the KdV equation,

ut +6uux + u3x =0. (3.72)

See Appendix C for proof of this assertion. Consider the following two examples which illustrate how to work with the linear integral equation (3.67). This will require using Cauchy’s Residue Theorem [67, 68],

0, if z0 is outside C, f(z) dz = 2πif(z0), if z0 is inside the interior to C, (3.73) C z z0 Z − πif(z0), if z0 is on C, where f is analytic in a simple connected domain D and C is a simple closed positively oriented (i.e., traversed counterclockwise) contour that lies within D.

26 Example 3.1.

To illustrate how (3.67) and (3.71) can be used to compute, for example, the solitary wave solution to (3.72), we follow [65] and choose as suitable measure

1 c1 dλ(k)= dk or dλ(k)= c1δ(k k1) dk, (3.74) 2πi k k1 − − where the delta function is centered at k1 = iκ1 and c1 and κ1 are arbitrary real constants.

The contour C surrounds the simple pole k1 (for the ﬁrst choice) or passes through k1 = iκ1 if a delta function is used.

Setting k = k1 and using (3.73) with the ﬁrst choice in (3.74), integral equation (3.67) simpliﬁes into an algebraic equation. Indeed,

1 c ϕ(ℓ) ϕ(k )+ iρ(k ) 1 dℓ (3.75) 1 1 2πi (ℓ k ) (ℓ + k ) ZC − 1 1 c1 = ϕ(k1)+ iρ(k1) ϕ(k1)= ρ(k1), (3.76) 2k1

where we suppressed the variables x and t in ϕ(x,t; k1),ϕ(x,t; ℓ), and ρ(x,t; k1). Solving (3.75), one gets

2k1ρ(k1) ϕ(k1)= . (3.77) 2k1 + ic1ρ(k1) Using (3.69) and the same measure,

1 c ϕ(k) 2k c ρ(k ) 2ik c ρ(x,t; k ) w(x,t)= 1 dk = c ϕ(k )= 1 1 1 = 1 1 1 (3.78) 2πi k k 1 1 2k + ic ρ(k ) −c ρ(x,t; k ) 2ik ZC − 1 1 1 1 1 1 − 1 and, from (3.71),

3 4ic1k1ρ(x,t; k1) u(x,t)= wx = c1∂xϕ(x,t; k1)= 2 . (3.79) − − c ρ(x,t; k ) 2ik 1 1 − 1 2K1δ1 Setting k1 = iκ1 =2iK1 and c1 =4K1e , where δ1,K1 are arbitrary real constants, after some algebra, (3.79) can be written as

2 2 u(x,t)=2K1 sech (Θ1), (3.80)

27 where Θ = K (x 4K2t δ ). This is the well-known [45, p. 83] solitary wave solution of 1 1 − 1 − 1 (3.72).

Example 3.2.

Pure N-soliton solutions for the KdV equation can be computed in a similar way by choosing

1 N c N dλ(k)= j dk or dλ(k)= c δ(k k ) dk, (3.81) 2πi k k j − j j=1 j j=1 X − X where the delta function is centered at kj = iκj, and cj,κj are arbitrary real constants.

As above, the contour C either surrounds the poles kj or, for the second choice, C passes

through all kj = iκj.

Setting k = kn (for one ﬁxed value of n at the time) and using any of the two measures above, (3.67) reduces to an algebraic system from which the N-soliton solution to (3.70) and (3.72) can be computed using

w(x,t)= cjϕ(kj) (3.82) j=1 X and (3.71), respectively. Indeed, for k = kn (with ﬁxed n) and the ﬁrst choice in (3.81),

(3.67) for ϕ(kn) becomes

1 N c ϕ(ℓ) ϕ(k )+ iρ(k ) j dℓ (3.83) n n 2πi (ℓ k ) (ℓ + k ) C j=1 j n Z X − 1 N c ϕ(ℓ) = ϕ(k )+ iρ(k ) j dℓ (3.84) n n 2πi (ℓ k ) (ℓ + k ) j=1 C j n X Z − N c = ϕ(k )+ iρ(k ) j ϕ(k )= ρ(k ). (3.85) n n k + k j n j=1 j n X Doing the same for each k with n =1, 2, ,N, yields the algebraic system. For example, n ··· for N = 2 one gets

28 c c ϕ(k )+ iρ(k ) 1 ϕ(k )+ 2 ϕ(k ) = ρ(k ), 1 1 2k 1 k + k 2 1 1 1 2 c c ϕ(k )+ iρ(k ) 1 ϕ(k )+ 2 ϕ(k ) = ρ(k ). (3.86) 2 2 k + k 1 2k 2 2 1 2 2 Solving this system yields

ic2(k1 k2) − ρ(k1) 1+ 2k (k +k ) ρ(k2) ϕ(k ) = 2 1 2 , 1 ic1 ic2 c1c2 (1 + ρ(k ))(1 + ρ(k )) + 2 ρ(k )ρ(k ) 2k1 1 2k2 2 (k1+k2) 1 2

ic1(k2 k1) − ρ(k2) 1+ 2k (k +k ) ρ(k1) ϕ(k ) = 1 1 2 . (3.87) 2 ic1 ic2 c1c2 (1 + ρ(k ))(1 + ρ(k )) + 2 ρ(k )ρ(k ) 2k1 1 2k2 2 (k1+k2) 1 2

ic1 ic2 c1c2a12 Note that the common denominator can be written as 1+ ρ(k1)+ ρ(k2) ρ(k1)ρ(k2), 2k1 2k2 − 4k1k2 where a = [(k k )/(k + k )]2 is a coupling coeﬃcient (cf. (3.127)). 12 1 − 2 1 2 With regard to (3.82),

ic1c2(k1+k2)a12 c1ρ(k1)+ c2ρ(k2)+ ρ(k1)ρ(k2) w(x,t)= c ϕ(k )+ c ϕ(k )= 2k1k2 . (3.88) 1 1 2 2 ic1 ic2 c1c2a12 1+ ρ(k1)+ ρ(k2) ρ(k1)ρ(k2) 2k1 2k2 − 4k1k2 This solution was reported in [69, p. 6] modulo a small misprint.

4K (K +K ) For the choice, k = iκ = 2iK and c = j 1 2 e2Kj δj , the 2-soliton solution of the j j j j K2 K1 − KdV equation (based on (3.88)) can be put in nice forms as

4(K2 K2)((K2 K2)+K2cosh(2Θ )+K2cosh(2Θ )) u(x,t)= w = 2 − 1 2 − 1 1 2 2 1 , (3.89) − x [(K K )cosh(Θ +Θ )+(K +K )cosh(Θ Θ )]2 2 − 1 1 2 2 1 2 − 1 K2sech2(Θ )+ K2csch2(Θ ) =2 K2 K2 1 1 2 2 , (3.90) 2 − 1 [K tanh(Θ ) K coth(Θ )]2 1 1 − 2 2 where Θ = K (x 4K2t δ ) with K ,δ (j =1, 2) arbitrary constants. Expression (3.89) j j − j − j j j can be found in [45, p. 83]. Expression (3.90) is given in [70].

29 3.3.1 Discrete Direct Linearization

Starting from the integral equation (3.67) for the KdV equation, Nijhoﬀ et al. [13] and Quispel et al. [30] derived the direct linearization formalism for a general class of P∆Es, including the discrete potential KdV and mKdV equations, as well as the discrete-time Toda equation and the lattice sine-Gordon equation. This requires a vector formulation of (3.67),

u (x,t) u (x,t)+ iρ (x,t) dλ(ℓ) ℓ = ρ (x,t)c , (3.91) k k k + ℓ k k ZC where, adhering to the literature on P∆Es, the dependence on the variable k is written as a subscript. As before, the contour C in the complex k-plane and measure dλ(k) are to be

chosen such that the solution uk(x,t) is unique for a given ρk(x,t).

Note that the scalar function ϕ(x,t; k) in (3.67) is replaced by an inﬁnite vector uk(x,t)

(j) (j) j with components u ,j integer, corresponding to a factor c k− of the inﬁnite vector c k k ≡ k

which multiplies ρk(x,t) on the right hand side of (3.67). Furthermore, ρk(x,t) can be taken more general than the plane-wave function (3.68) used in the direct linearization formulation of the KdV equation.

The goal is to ﬁnd uk(x,t) as a function of the complex variable k and to obtain solutions of nonlinear PDEs by using (3.69). The matrix analog [30] of (3.69) then becomes

U = i dλ(k)ukck, (3.92) ZC where U is a (symmetric) dyadic matrix obtained by integration of the dyadic ukck over the same contour and with respect to the same measure used in (3.91). The full vector formalism (which leads to inﬁnite matrices) is not needed to explain the key idea behind the transition from the continuous to discrete scenario. To keep matters as transparent as possible, we restrict ourselves to the case j = 0 leading to a scalar formalism where u u(0), c(0) =1, and w = u , i.e., the (0, 0)-element of U. k ≡ k k 0,0

30 In essence, Capel and his collaborators [13, 30] introduced two Bäcklund transformations: The first Bäcklund transformation comes from using

p + k ρ(1) = ρ , (3.93) k p k k − ! (1) (with complex parameter p) in (3.67) leading to a ﬁrst solution uk . In turn, (3.69) will then lead to a solution w(1)(x,t). Likewise, the second B¨acklund transformation,

q + k ρ(2) = ρ , (3.94) k q k k − ! (with parameter q) results in a second solution w(2)(x,t). Note that, e.g., (3.93) is equivalent to introducing a singular transformation of the mea-

p+k sure, dλ(k) p k dλ(k), assuring that the solution of (3.91) remains unique and using a 7→ − contour that does not pass through p and p. − As shown in Section 3.2, by combining both B¨acklund transformations and applying the Bianchi Permutability Theorem, one gets

(p + q + w w(12))(p q + w(2) w(1))= p2 q2. (3.95) − − − − We now turn to the discrete formulation. The equivalence of the Bianchi identity and the corresponding P∆E can be established by a straightforward discretization of the plane-wave

factor. Indeed, it suﬃces to replace (3.68) by a function ρk(n,m) of the lattices variables n

and m such that ρk(n,m) satisﬁes the linear diﬀerence equations,

p + k ρ (n +1,m)= ρ (n,m), (3.96) k p k k − ! and

q + k ρ (n,m +1) = ρ (n,m), (3.97) k q k k − ! which are both satisﬁed by

n m p + k q + k ρ (n,m)= ρ (0, 0). (3.98) k p k q k k − ! − !

31 (1) (2) 7 Obviously, ρk = ρk(n +1,m) and ρk = ρk(n,m + 1). Quispel [69] has shown that (3.95) can be replaced by

(p + q + x x )(p q + x x )= p2 q2, (3.99) − 12 − 2 − 1 − where x = wn,m, i.e., the discrete analog to the continuous solution w of the potential KdV equation. As shown in Appendix B, equation (3.99) is equivalent to the discrete pKdV equation (3.65) after the replacement

x x + np + mq + c, (3.100) 7→ where c is an arbitrary constant. This shows how nonlinear P∆Es naturally arise from B¨acklund transformations applied to the discrete version of the direct linearization formula. Alternatively, they are obtained from that formula by replacing the continuous plane-wave factor by a proper discretization. Furthermore, N-soliton solutions of the lattice potential KdV equation (3.99) can be obtained [69] using the measures given in (3.81). For example, the 2-soliton solution follows

from (3.88) by replacing ρ(k1)= ρ(x,t; k1) and ρ(k2)= ρ(x,t; k2) by ρk1 (n,m) and ρk2 (n,m) using (3.98). In [30], the authors prove a theorem which leads to the so-called (α, β)-lattice as given in Table 3.3.

Theorem 1. Let uk(n,m) be a solution of the singular linear integral equation, u (m,n) ρ (m,n) u (n,m)+ iρ (m,n) ℓ dλ(ℓ)= k , (3.101) k k k + ℓ k + α ZC where α is an arbitrary complex parameter, ρk(n,m) as in (3.98), and contour C and measure dλ(k) as speciﬁed above. Then,

u (n,m) u(n,m)= i k dλ(k), (3.102) k + β ZC

7In [69, p. 6-7], the meaning of p and q is reversed. Furthermore, to obtain equations with real coeﬃcients, Quispel’s variable u needs to be replaced by iu, as indicated in (3.92).

32 where β is an arbitrary complex parameter, satisﬁes the nonlinear P∆E,

[(p α)x (p + β)x + 1][(p β)x (p + α)x + 1] − − 1 − 2 − 12 = [(q α)x (q + β)x + 1][(q β)x (q + α)x + 1], (3.103) − − 2 − 1 − 12 where α,β,p and q are ﬁxed. For simplicity of notation, x = un,m and x1 = un+1,m, x2 = un,m+1 and x12 = un+1,m+1 denote its shifts.

Proof. The proof is given in [30].

3.4 Bilinear Operator

In 1971 Hirota [71] proposed a direct method for constructing multisoliton solutions of integrable nonlinear evolution equations. As a byproduct of his method, he also introduced a technique for constructing discrete analogues of many completely integrable PDEs, including the KdV, mKdV, sine-Gordon, and NLS equations. Indeed, in a series of five seminal papers [5–9], Hirota presented a new approach to obtain and solve integrable nonlinear partial difference equations using a discrete version of the bilinear operator. We will first discuss Hirota’s direct method [72] for nonlinear PDEs. Hirota’s method consists of several steps, each of which requires some creativity:

First, introduce a change of dependent variable(s) to transform the given PDE into • an equivalent PDE (or system of PDEs) that is homogeneous in the new dependent variable(s).

Second, introduce suitable bilinear operators and write the homogeneous PDE (or • PDEs) in bilinear form.

Then, use a formal perturbation expansion to solve the bilinear equation(s). As shown • in, e.g., [48] in the case of soliton solutions, this formal series truncates. Speciﬁcally, for the KdV equation, if the original equation admits an N soliton solution, this series −

33 will truncate at level n = N provided f1 (the ﬁrst non-constant term in the expansion) is the sum of N exponential terms.

Finally, use mathematical induction (or explicit veriﬁcation) to show that the derived • (formal) soliton solution is correct.

We now illustrate these steps for the KdV equation. Motivated by the form of the N soliton − solution [37] for

ut +6uux + uxxx =0, (3.104) and by the Cole-Hopf transformation which linearizes the Burgers equation8 Hirota introduced the transformation,

ff f 2 u(x,t)=2∂2 ln f(x,t)=2 xx − x , (3.105) x f 2 where f(x,t) is a new dependent variable. Applying this transformation to (3.104), after one integration with respect to x, (3.104) can be replaced by

ff f f + ff 4f f +3f 2 =0. (3.106) xt − x t 4x − x 3x 2x Note that (3.104) has two linear terms and one quadratic term, whereas (3.106), though more complicated in appearance, is quadratic in f and its derivatives. The transformed equation (3.106) can be rewritten in terms of Hirota’s bilinear operators,

Dt and Dx, resulting in the bilinear form of (3.104),

D D + D4 (f f)=0. (3.107) x t x · The bilinear operators, Dt and Dx, are deﬁned on pairs of functions f(x,t) and g(x,t) as follows

m n m n ∂ ∂ ∂ ∂ D D (f g)= f(x,t)g(x′,t′) , (3.108) x t ′ · ∂x − ∂x′ ∂t − ∂t′ x =x t′=t

8 The Burgers equation, ut 2uux uxx = 0, can be transformed into the heat equation, ft = fxx, with the − − 2fx Cole-Hopf transformation u = 2(ln f(x,t))x = f .

34 with non-negative integers m and n. The details of the derivation of (3.107), together with equivalent deﬁnitions and properties of these bilinear operators are shown in Appendix D. It is worth noting that this bilinearization process is far from algorithmic. In fact, it is not obvious what type of transformation is needed or how many dependent variables should be

2 introduced. For example, the modiﬁed KdV equation, ut +6u ux +u3x =0, can no longer be

f bilinearized with a single function f. Among several choices [72, p. 45], for example, u = g , leads to two coupled bilinear equations [72, p. 40]. We continue with (3.106) involving only one new dependent variable (f). Hirota’s direct method continues by applying a perturbation-like method to (3.107). Assuming a formal series of the solution

∞ f =1+ εf + ε2f + ε3f + =1+ εnf , (3.109) 1 2 3 ··· n n=1 X 9 where ε serves as a bookkeeping parameter and f1(x,t), f2(x,t), etc., are unknown functions, which, when (3.109) is substituted into (3.107), must satisfy a system of equations arising at separate powers of ε. This system of equations

ε : D D + D4 (f 1+1 f )=0, (3.110) x t x 1 · · 1 ε2 : D D + D4 (f 1+ f f +1 f )=0, (3.111) x t x 2 · 1 · 1 · 2 ε3 : D D + D4 (f 1+ f f + f f +1 f )=0, (3.112) x t x 3 · 2 · 1 1 · 2 · 3 ε4 : D D + D4 (f 1+ f f + f f + f f +1 f )=0, (3.113) x t x 4 · 3 · 1 2 · 2 1 · 3 · 4 . . . . n n 4 ε : DxDt + Dx fi fn i =0, with f0 =1, (3.114) · − i=0 ! . . X . .

terminates at level n = N, i.e., f = 0 for i N, where N is the number of exponential i ≥

terms taken in f1. Explicit formulas for soliton solutions may then be computed using the properties of Hirota’s bilinear operation (see Appendix D).

9The parameter ε> 0 is not necessarily small as in traditional perturbation methods.

35 To illustrate Hirota’s method let us ﬁrst consider the case N = 1, which will yield the well-known solitary wave (1-soliton) solution of (3.104). The simpliﬁcation of (3.110),

4 (∂x∂t + ∂x)f1 =0, (3.115) admits the following simple solution,

f := eη1 , η = k x ω t + δ with constants k , ω ,δ , (3.116) 1 1 1 − 1 1 1 1 1 which will lead to a 1-soliton solution. Then (3.115) determines the nonlinear dispersion

3 relation, ω1 = k1. Substituting (3.116) into (3.111) gives

2(∂ ∂ + ∂4)f = (D D + D4)(f f )=0. (3.117) x t x 2 − x t x 1 · 1

Thus, f2 = 0 may be chosen. Subsequent substitutions into (3.112), (3.113), etc., allow f 0 for i 2, to be chosen. Thus, (3.109) reduces to i ≡ ≥ f =1+ eη1 (3.118) for ε = 1. It then follows from (3.105), after some algebra, that

1 1 u(x,t)= k2 sech2 (k x k3t + δ ) , (3.119) 2 1 2 1 − 1 1 or, after setting k1 =2K1 and δ1 = 2∆1,

u(x,t)= K2 sech2(K x 4K3t +∆ ), (3.120) 1 1 − 1 1 which is the long-known solitary wave solution of the KdV equation. For a two-soliton solution of the KdV equation take

f := eη1 + eη2 , η = k x ω t + δ , i =1, 2, (3.121) 1 i i − i i as a solution to (3.110) and proceed as before. Substitution of f1 into (3.111) yields

1 (∂ ∂ + ∂4)f = D D + D4 (f f ), (3.122) x t x 2 − 2 x t x 1 · 1 1 = D D + D4 ((eη1 + eη2 ) (eη1 + eη2 )) , (3.123) − 2 x t x · = (k k )(ω ω )+(k k )4 eη1+η2 , (3.124) − 1 − 2 2 − 1 1 − 2 =3k k (k k )2eη1+η2 , (3.125) 1 2 1 − 2

36 3 using the dispersion law ωi = ki (i =1, 2). Thus, a solution to this equation would be of the form

η1+η2 f2 = a12e , (3.126) where a12 is an unknown constant. Substitution of (3.126) in the left hand side of (3.122), after some additional algebra, gives

k k 2 a = 1 − 2 . (3.127) 12 k + k 1 2 As before, substitution into the subsequent perturbation equations allows for the conclusion that f = 0 for i 3. Hence, i ≥

η1 η2 2 η1+η2 f =1+ ε(e + e )+ ε a12e , (3.128) which can then be written as

η1 η2 η1+η2 f =1+ e + e + a12e , (3.129) provided ε = 1. Returning to (3.105) yields the two-soliton solution for the KdV equation. The computation of the three-soliton solution proceeds along similar lines. Choosing

η1 η2 η3 f1 = e + e + e , (3.130) eventually yields

η1 η2 η3 η1+η2 η1+η3 η2+η3 η1+η2+η3 f =1+ e + e + e + a12e + a13e + a23e + a12a13a23e , (3.131)

k k 2 where a = i − j with i, j =1, 2, 3 (i

N ηi f1 = e , (3.132) i=1 X

37 an exact solution [73] of (3.107) can be expressed as

N N

f = exp Aijµiµj + µiηi , (3.133)

µi=0,1 " i

in (3.133), the ﬁrst one is over all combinations of µ1 = 0, 1, µ2 = 0, 1, ..., µN = 0, 1; the second over all possible pairs (i, j) chosen from the set 1, 2,...,N , under the condition { } 1 i

3.4.1 Discrete Hirota Formalism

In 1977, Hirota developed an analogous discrete method [5–9] to ﬁnd fully discrete versions of well-known completely integrable PDEs. To do so, he deﬁned a discrete operator,

Dn, as follows

∂ ∂ exp(δDn)(f(n) g(n)) = exp δ( ) (f(n)g(n′)) = f(n + δ)g(n δ), (3.134) · ∂n − ∂n ′ − ′ ! n =n

where δ is assumed to be a lattice parameter, i.e., δ is the distance between lattice points n and n + 1. As shown in Appendix D, operator exp(δDn) serves as the discrete analogue of

Dx. Focused on the KdV equation, Hirota [74, 75] considered a diﬀerential-diﬀerence form10 of (3.104),

d wn = wn 1 wn+ 1 . (3.135) dt 1+ w − 2 − 2 n Next, he showed [5] that the transformation

10 d wn Hietarinta et al. [64] treat = wn−2 wn+2, obtained from (3.135) by a scaling of n. dt 1+wn − 38 fn+1(t)fn 1(t) − wn = 2 1 (3.136) fn(t) − cosh 1 D (f (t) f (t)) 2 n n · n = 2 1, (3.137) fn(t) − where t is treated as a parameter, allows one to derive the bilinear form11 of (3.135),

1 1 1 1 sinh( D ) sinh( D ) D +2sinh( D ) (f f ) cosh( D )(f f ) =0, (3.138) 4 n 4 n t 2 n n · n 4 n n · n which is satisﬁed if

1 1 sinh( D ) D + 2sinh( D ) (f (t) f (t))=0. (3.139) 4 n t 2 n n · n The latter equation is the bilinear form of the differential-difference KdV equation (3.135). The next step is to get to a fully discrete version of (3.104), i.e., a nonlinear P∆E corresponding to the KdV equation. Based on a comparison of the bilinear forms, (3.107) and (3.139), for the continuous and differential-difference versions of the KdV equation, respectively, Hirota proposed a bilinear form for a difference-difference analogue of the KdV equation,

1 δ 1 2 δ 1 sinh D + D sinh D + 2sinh( D ) f (t) f (t) =0. (3.140) 2 2 t 2 n δ 2 t 2 n n · n The proposed form – which was crucial for the derivation – makes sense for the following reasons: (i) the expression in the square brackets is symmetric with respect to the space

2 δ and time coordinates, and (ii) the limits make sense. Indeed, limδ 0 sinh Dt = Dt and → δ 2 1 1 limδ 0 sinh (δDt + Dn) = sinh Dn . Therefore, both limits match the corresponding → 4 4 terms in (3.139).

11The derivation of (3.138) was veriﬁed as part of the research for this dissertation. Due to its length it will not be shown here.

39 ˜ 1 ˜ δ To complete the path from (3.140) to a P∆E, set Dn = 4 Dn and Dm = 4 Dt, and identify t with the discrete variable m. Then (3.140) can be written as

2 sinh D˜ + D˜ sinh 2D˜ + 2sinh 2D˜ f f =0, (3.141) n m δ m n n,m · n,m treating t as a second discrete variable (m). Expressing sinh and cosh in terms of exponentials (see Appendix D) and using formula (3.134) repeatedly, yields

1 1 fn+3,m+1fn 3,m 1 + fn+1,m+3fn 1,m 3 1+ fn+1,m 1fn 1,m+1 =0. (3.142) − − δ − − − δ − − Finally, introducing

fn+2,mfn 2,m Un,m = − 1, (3.143) fn,m+2fn,m 2 − − it can be shown [64, p. 230] that if fn,m satisﬁes (3.142) then Un,m satisﬁes 1 1 δ(Un+2,m Un 2,m)= , (3.144) − − 1+ Un,m+2 − 1+ Un,m 2 − which can be easily transformed into

2 2 1 1 un,m+1 un+1,m =(p q ) , (3.145) − − un,m − un+1,m+1 ! using the transformation, U u 1 wheren ˜ = n + 1 (2 i j) andm ˜ = m + n+i,m+j 7→ n,˜ m˜ − 4 − − 1 1 4 (2 + i j), and where δ = p2 q2 . From the bilinear form (3.140) Hirota also constructed − − a difference-difference analogue for the KdV equation by reversing the process. By analogy, reversal in the continuous case would amount to getting back to (3.104) from (3.107). To do so, Hirota defined the transformation [5],

1 cosh( 2 Dn)(fn fn) Wn(t)= · 1, (3.146) cosh( δ D )(f f ) − 2 t n · n fn+ 1 (t)fn 1 (t) = 2 − 2 1, (3.147) f t + δ f t δ − n 2 n − 2

40 which yields (after some algebra)

Wn(t) ∆t = Wn 1 (t) Wn+ 1 (t), (3.148) 1+ Wn(t)! − 2 − 2

where ∆t operating on, say, Fn(t) is defined by 2 δ 1 δ δ ∆ F (t)= sinh ∂ F (t)= F (t + ) F (t ) . (3.149) t n δ 2 t n δ n 2 − n − 2 In summary, (3.148) is a difference-difference analogue of the KdV equation which can be shown to reduce to the discrete KdV equation (3.135) in the limit for δ 0. → A more straightforward approach to discretization was recently proposed by Hirota [76]. The starting point is a differential-difference equation of a completely integrable PDE, e.g.,

dfn(t) 2 +(1+ fn)(fn+1 fn 1)=0, (3.150) dt − − which is a space-discrete analogue of the mKdV equation. Without going into full details, the time-derivative is then replaced by the forward diﬀerence,

df (t) 1 n (f f ), (3.151) dt → δ n,m+1 − n,m leading to a discrete analogue of the bilinear operator Dt (deﬁned in (3.108)),

1 D (f (t) g (t)) = g′ (t)f (t) g (t)f ′ (t) (g f g f ). (3.152) t n · n n n − n n → δ n,m+1 n,m − n,m n,m+1

Doing so, Hirota [76] has shown that the two coupled equations,

2 vn,m+1 vn,m + δ(1 + vn,m)wn,m(vn+1,m vn 1,m+1)=0, − − − 2 wn+1,m 1+ vn,m = 2 , (3.153) wn,m 1+ vn,m+1 constitute the difference-difference version of the mKdV equation. Unfortunately, many of Hirota’s difference-difference equations (like (3.153)) are neither defined on the quadrilateral nor consistent around the cube. They are therefore outside the scope of this dissertation.

41 3.5 Multidimensional Consistency

The fundamental characterization of integrable P∆Es as being multi-dimensionally consistent [16, 38] is intimately related to the existence of a Lax pair. The concept of embedding the partial difference equation into a 2D lattice was formulated in [54] and then proposed as an integrability test in [15, 38]. Since the introduction, the concept of multi-dimensional consistency (i.e., consistency around the cube) has been widely used as a definition of integrability. It is the extension of this 2D definition to the definition of 3D consistency that forms the basis for the software implementation that accompanies this dissertation. Lax pairs for P∆Es are not only crucial for applying the IST, they can also be used to construct integrals for mappings and correspondences obtained as periodic reductions, using the so-called staircase method. This method was developed in [39] and extended in [40] to cover more general reductions. Essential to the staircase method is the construction of a product of Lax matrices (the monodromy matrix) whose characteristic polynomial is an invariant of the evolution. In fact, the monodromy matrix can be interpreted as one of the Lax matrices for the reduced mapping [41–43]. Through expansion of the characteristic equation of the monodromy matrix in the spectral parameters a number of functionally independent invariants can be obtained. A recent investigation [44] supports the idea that the staircase method provides sufficiently many integrals for the periodic reductions to be completely integrable (in the sense of Liouville-Arnold).

3.5.1 Consistency Around the Cube

As mentioned in Chapter 1 the general idea is to embed a nonlinear P∆E,

(x,x ,x ,x ; p,q)=0. (3.154) F 1 2 12 consistently into a multi-dimensional lattice by imposing copies of the same equation, albeit with different lattice parameters in different directions. More specifically, as shown in Fig- ure 1.1, the field variable x = xn,m depends on lattice variables n and m. A shift of x in the horizontal direction (the 1 direction) is denoted by x x . A shift in the vertical or − 1 ≡ n+1,m

42 2 direction by x x and a shift in both directions by x x . Furthermore, − 2 ≡ n,m+1 12 ≡ n+1,m+1 F depends on the lattice parameters p and q which correspond to the edges of the quadrilateral.

It is also assumed that the initial values for x, x1 and x2 can be specified and that the value of x12 can be uniquely determined by (3.154). To have single-valued maps, we assume that is multi-affine [15], or, as sometimes F referred, multi-linear. Interestingly, Atkinson [77] and Atkinson & Nieszporksi [78] have recently given examples of P∆Es that are multi-quadratic and multi-dimensionally consistent. In the simplest case, is a scalar relation between values of a single dependent variable F x and its shifts (located at the vertices of an elementary square). As discussed in Section 3.2 Nonlinear lattice equations of type (3.154) arise, for example, as the permutability condition for Bäcklund transformations associated with integrable partial differential equations (PDEs). In more complicated cases, is a nonlinear vector function of the vector x with several F components. In that case, (3.154) represents a system of P∆Es which are called multi- component lattice equations. In such systems some equations might only be defined on singles edges of the square, multiple adjacent edges or on the whole square. The algorithm for deriving a corresponding Lax pair for either the scalar or vector P∆E is discussed in Chapter 5. To arrive at a cube, the planar quadrilateral is extended into the third dimension as shown in Figure 3.3, where k is the lattice parameter in the third direction. Although not explicitly shown in the figure, all parallel edges carry the same lattice parameters. A key assumption is that the original equation(s) holds on all faces of the cube. These equations can therefore be generated by changes of variables and parameters, or shifts of the original P∆E. On the cube, they can be visualized as either translations, or rotations of the faces. For example, the equation on the left face can be obtained via a rotation of the front

face along the vertical axis connecting x and x2. This amounts to applying to (3.154) the

43 x23 x123

x2 x12

q x3 x13 k

x p x1

Figure 3.3: The P∆E is deﬁned on the cube.

substitutions

x x , x x , and p k, (3.155) 1 → 3 12 → 23 → yielding (x,x ,x ,x ; k,q) = 0. The equation on the back face of the cube can be generated F 3 2 23 via a shift of (3.154) in the third direction, letting

x x , x x , x x , and x x , (3.156) → 3 1 → 13 2 → 23 12 → 123 which yields (x ,x ,x ,x ; p,q)=0. F 3 13 23 123

The equations on the back, right, and top faces of the cube all involve the unknown x123.

Solving them yields three expressions for x123. Consistency around the cube of the P∆E

requires that one can uniquely determine x123 and that all three expressions coincide. As discussed in [31], this three-dimensional consistency establishes integrability. The consistency property does not depend on the actual mappings used to generate the P∆Es on the various faces of the cube. Mappings such as (3.155) and (3.156), which express the symmetries of the P∆Es are merely a tool for generating the needed P∆Es quickly.

3.5.2 ABS Classiﬁcation

In [15] Bobenko and Suris showed that integrability for two-dimensional systems follow from three-dimensional consistency. Then in [16] Adler, Bobenko and Suris extended the consistency property and showed that it provided an eﬀective tool for ﬁnding and classifying

44 integrable systems in certain classes of equations – specifically integrable equations defined on quad-graphs. As outlined in [16], the Adler, Bobenko and Suris (ABS) classification included additional assumptions.

It is assumed that (3.154) can be uniquely solved for any one of its arguments x, x , • 1

x2 or x12 which leads to the assumption of linearity. Thus, (3.154) is assumed to be aﬃne linear, i.e., linear in each argument:

(x,x ,x ,x ; p,q)= a (xx x x )+ a (x x x )+ a (xx x ) + a , (3.157) F 1 2 12 1 1 2 12 2 1 2 12 3 2 12 ··· 16

for some constants a1, a2,..., a16.

As (3.154) is deﬁned on the quadrilateral and no preferences are given to any individual • variable involved, symmetry is assumed. Thus, it is assumed that the equation is

invariant under D4, the dihedral group of symmetries of the square:

(x,x ,x ,x ; p,q)= ε (x,x ,x ,x ; p,q)= δ (x ,x,x ,x ; p,q), (3.158) F 1 2 12 F 2 1 12 F 1 12 2 for ε,δ = 1. ±

An additional assumption was found to hold in all known nontrivial examples. It is •

assumed that (3.154) has the tetrahedron property. That is, when solving for x123, as part of the CAC property, the resulting function does not depend on x, as visually represented in Figure 3.4.

These criteria led to the ABS-classiﬁcation of integrable nonlinear lattice equations [16, 79]. Interestingly, in addition to the criteria given above, Adler et al. found that the equations of the ABS classiﬁcation possessed another property closely related to the tetrahedron property. For each of these equations, there exists a three-leg form of (3.154),

ψ(x,x ; p) ψ(x,x ; q)= ϕ(x,x ; p,q), (3.159) 1 − 2 12

45 x23 x123

x2 x12

q x3 x k 13

x p x1

Figure 3.4: The tetrahedron property.

for some ψ and ϕ. For example, the H1 equation (using the ABS classiﬁcation nomenclature),

(x x )(x x )+ q p =0, (3.160) − 12 1 − 2 − can be written as (3.159) with α β ψ(u,v; α)= u + v, and ϕ(u,v; α, β)= − . (3.161) u v − Whereas, for the Q1 equation (with δ = 0),

p(x x )(x x ) q(x x )(x x )=0, (3.162) − 2 1 − 12 − − 1 2 − 12 the three-leg form is given by α α β ψ(u,v; α)= , and ϕ(u,v; α, β)= ψ(u,v; α β)= − . (3.163) u v − u v − − This additional property is not an additional criteria for integrability. Instead, the authors showed that the tetrahedron property is a necessary condition for the existence of the three- leg form. However, the three-leg form was used for further investigation into the Lagrangian structures for the equations in the ABS classiﬁcation (see Appendix B).

The Q1 - Q3, A1 - A2, H1 - H3 equations, as given in the ABS classiﬁcation are presented in Table 3.2 which includes Lax pairs as calculated by the software accompanying this dissertation. The Q4 equation is also included in the ABS classiﬁcation. This equation, also known as the Adler-Krichever-Novikov lattice system [80], has been excluded from Table 3.2 as its

46 complexity has proven to be beyond the capabilities of the software. However, a Lax pair for the Q4 equation was constructed in [38]. Based on [80], the equation and a corresponding Lax pair is included in Table 3.1 for completeness12.

Table 3.1: The Q4 equation

Adler’s form, as parameterized by elliptic functions (i.e., (p, A) = ℘(ρ),℘′(ρ) and (q,B) = ℘(σ),℘′(σ) where ℘ is the Weierstrass elliptic function (or ℘-function), for constantsρ and σ), is given by

α xx x x α (xx x + x x x + xx x + xx x )+ α (xx + x x ) 0 1 2 12 − 1 1 2 1 2 12 2 12 1 12 2 12 1 2 α (xx + x x ) α (xx + x x )+ α (x + x + x + x )+ α = 0 − 3 1 2 12 − 4 2 1 12 5 1 2 12 6

where the coeﬃcients αi(p,q) are given by,

2 2 α0 = A + B, α1 = Aq + Bp, α2 = Aq + Bp , AB(A + B) g AB(A + B) g α = Aq2 + B 2p2 2 , α = Bp2 + A 2q2 2 , 3 2(q p) − − 4 4 2(p q) − − 4 − − g g g2 α = 3 α + 2 α , α = 2 α + g α , 5 2 0 4 1 6 16 0 3 1

and where A2 = r(p), B2 = r(q), r(x) = 4x2 g x g . − 2 − 3

A corresponding Lax pair (i.e., the L matrix) is given as

1 α xx α x + α x α α xx α (x + x ) α L = 1 1 − 2 1 4 − 5 3 1 − 5 1 − 6 , D " α0xx1 α1(x + x1) α3 α1xx1 + α2x α4x1 + α5# − − − − where D = (a k)2CEH(x,x ,p) with the spectral parameter k also parameterized by − 1 the elliptic function (i.e., (k,C)= ℘(τ),℘ (τ) ), E = ℘(τ ρ) and with p ′ − g 2 H(x, y, z) := xy + xz + yz + 2 (x + y +z)(4xyz g ). 4 − − 3

12In addition to Adler’s form given here, Nijhoﬀ [38] and Hietarinta [81] have derived alternate forms of the Q4 equation.

47 Table 3.2: Scalar P∆Es of ABS classiﬁcation

Name Equation Matrix L Alternate t values Ref.

x p k xx1 H1 (x x12)(x1 x2)+ q p =0 t − − with t =1 [16] − − − 1 x1 " − # (Also known as lpKdV.)

2 2 p k + x p k +(p k)(x + x1) xx1 t − − − − " 1 (p k + x1) # H2 (x x12)(x1 x2)+(q p)(x + x1 + x2 + x12 + p + q)=0 − − [16] − − − 1 with t = √p + x + x1

2 2 kx δ(p k ) pxx1 t − − − 2 2 " p kx1 # H3 p(xx1 + x2x12) q(xx2 + x1x12)+ δ(p q )=0 − [16] − − 1 with t = √δp + xx1 (For δ = 0, also known as lmKdV.)

2 kx + ηx1 p(xx1 + δ kη) t − where η = k p, 2 " p (kx1 + ηx) # − A1 p(x+ x2)(x1 +x12) q(x +x1)(x2 + x12) δ pq(p q)=0 − [16] − − − 1 with t = (δp (x + x1))(δp +(x + x1)) − p

2 2 2 (q p )(xx1x2x12 +1)+ q(p 1)(xx2 + x1x12) kγx (τ + pσxx1) 2 A2 − − t − where γ = p 1, [16] 2 pσ + τxx kγx − p(q 1)(xx1 + x2x12)=0 " 1 1 # σ = k2 1, and − − − 1 τ = p2 − k2. with t = − (xx1 p)(pxx1 1) − − p

48 Table 3.2: Continued.

Name Equation Matrix L Alternate t values Ref.

2 kx ηx1 p(xx1 δ kη) t − − − where η = k p, 1 p(x x2)(x1 x12) q(x x1)(x2 x12) " p (kx1 ηx) # − Q1 − − − − − − − t = [16] + δ2pq(p q)=0 1 δp (x x1) − with t = ± − (δp (x x1))(δp +(x x1)) − − − − (For δ = 0, also known as lsKdV.) p

η(kp x1)+ kx ℓ12 t − " p (η(kp x)+ kx1)# p(x x2)(x1 x12) q(x x1)(x2 x12) − − − − − − − where Q2 +pq(p q)(x + x1 + x2+x12) 2 [16] − ℓ12 = p[kη(kη + p x x1)+ xx1], η = k p − − − − pq(p q)(p2 pq + q2)=0 − − − 1 with t = 2 2 4 (x x1) 2p (x + x1)+ p − − p

2 2 2 2 2 4kp(σpx + τx1) γ(δ στ 4k pxx1) (q p )(xx12+x1x2)+ q(p 1)(xx1 + x2x12) t − − − when δ = 0, − − 4k2pγ 4kp(σpx + τx) 1 2 " 1 # t = or p(q 1)(xx2 + x1x12) − Q3 − − where γ = p2 1, σ = k2 1 and τ = p2 k2, px x1 [16] 2 −1 δ 2 2 2 2 − 1 − − (p q )(p 1)(q 1)=0 with t = t = . − 4pq − − − 2 2 px1 x 4p(px x1)(px1 x) δ γ − − − − p

Table 3.3: Additional Scalar P∆Es

Name Equation Matrix L Alternate t values Ref.

f(p)x τx1 f(k)xx1 t − − g(k) ((g(p)x1 τx) 1 " − − # (α,β) t = , − ((p α)x (p + β)x1)((p β)x2 (p + α)x12) where f(x):=(x α)(x β), (α p)x +(β + p)x1 lattice − − − − or − [13] g(x):=(x + α)(x−+ β), and− τ = p2 k2 ((q α)x (q + β)x2)((q β)x1 (q + α)x12)=0 1 − − − − − with − t = 1 (β p)x +(α + p)x1 t = − ((β p)x +(α + p)x1)((α p)x +(β + p)x1) − − p

49 CHAPTER 4 CONTINUUM LIMIT

In Chapter 3, the historical derivation of several P∆Es using a variety of techniques has been given. For example, the lpKdV equation (3.65) was derived using Bäcklund transformations, and using the direct linearization approach. Similarly, the lattice KdV equation (3.145) was derived using Hirota’s bilinear operator, as well as using Bäcklund transformations along with discrete variations. In addition to these equations, many more P∆Es, both scalar equations and systems of equations have been derived and will be presented (see Table 3.2 through Table 6.4). However, prior to discussing these equations further, the first logical step is to address the recovery of the corresponding PDE from a given P∆E. This recovery is done via a continuum limit. As the process begins with a discrete two-dimensional P∆E and progresses to a continuous PDE, one should not expect a unique result. In actuality, it has been shown [82] that the continuum limit of a given P∆E can generate an infinite family of continuous PDEs. However, it is essential to apply a continuum limit that retains the integrability property. In other words, if the P∆E is integrable, the limits should be taken so that the resulting PDE remains (completely) integrable.

4.1 Semi-Discrete Limits

Recall that the P∆E can be seen as existing on a two-dimensional lattice. Also recall that the Bäcklund transformations have been associated with translations on a two-dimensional lattice. Thus the Bäcklund parameters correspond to the respective widths on the quadrilateral lattice. Then, these parameters may be manipulated as with finite-difference approx- imations to differential equations, to determine the corresponding continuous equation. The recovery process can be described as “compressing” the lattice in two directions using Taylor

50 series expansions on the corresponding shift operators in both directions of the lattice. In this case, the Taylor expansion for a shift operator, ζ, is of the form,

h dy h2 d2y ζ(x) := y(x + h)= y(x)+ + + . (4.1) 1! dx 2! dx2 ··· In practice, this lattice compression may be done via a variety of limits. For the purpose of this discussion, we will proceed as outlined in [64] by performing the lattice compression in two steps, corresponding to the two Bäcklund parameters. The first step of the continuum limit yields a differential-difference equation – an equation with both continuous and discrete variables. The second and final step then yields a fully continuous equation. Thus, each step can be viewed as a semi-discrete (or semi-continuous) limit. Consider for example, the pKdV equation (3.99)

(p + q u + u )(p q + u u )= p2 q2. (4.2) − n+1,m+1 n,m − n,m+1 − n+1,m −

In [64], the authors linearized (4.2) by setting un,m = ερn,m and taking the terms linear in ε (ignoring terms in ε2) giving

(p + q)(ρ ρ )=(p q)(ρ ρ ), (4.3) n,m+1 − n+1,m − n+1,m+1 − n,m with corresponding solution (see Appendix A),

p k n q k m ρ = − − ρ , (4.4) n,m p + k q + k 0,0 where ρ0,0 is an arbitrary, constant initial solution. It is the nature of this solution that drives the type of continuum limit to use going forward. Indeed, the goal is to take the limits in such a way that when applied to the discrete solution (4.4) one gets exponential factors. Then the same limit could be applied to the corresponding nonlinear discrete equation (4.2) in order to derive its continuous counterpart.

4.2 Straight Continuum Limit

Recall that y n lim 1+ = ey, (4.5) n n →∞

51 for any real y. Consider, by deﬁning m := ξq, for some ﬁxed ξ,

q k m q k ξq ( 2k) ξq − = − = 1+ − . (4.6) q + k q + k q + k Note that k in the denominator becomes negligible as q . Thus, → ∞ m ξq q k ( 2k) 2kξ lim − = lim 1+ − = e− . (4.7) q q + k q q + k →∞ →∞ Therefore this is an appropriate limit in the first step of the continuum limit process. To apply this limit to (4.2), rewrite the discrete variable as, for fixed n, m u = µ (ξ) where ξ = ξ + , (4.8) n,m n 0 q for some fixed initial value ξ . Thus progression over the lattice as m m + 1 implies 0 7→ m +1 1 that ξ ξ + = ξ + . As such, the “compression” occurs along the lattice edge 7→ 0 q q q corresponding to the parameter associated with the Bäcklund transformation, ρ ρ , n,m 7−→ n,m+1 as depicted in Figure 4.1.

ρn,m+1 ρn+1,m+1

q p ρn,m ρn+1,m

Figure 4.1: Straight limit

Writing (4.2) in terms of the shift operator µ gives

1 1 p + q µ ξ + + µ (ξ) p q + µ ξ + µ (ξ) = p2 q2. (4.9) − n+1 q n − n q − n+1 − Applying a Taylor expansion then gives

1 p2 q2 = p + q µ + ∂ µ + + µ − − n+1 q ξ n+1 ··· n (4.10) 1 p q + µ + ∂ µ + µ , × − n q ξ n ··· − n+1

52 where µn denotes µn(ξ) and µn+1 denotes µn+1(ξ). Expanding, one gets

0= (p + q)[µ + ...] (p + q)µ (p q)[µ + ...] n − n+1 − − n+1 [µ + ...][µ + ...]+ µ [µ + ...]+ µ (p q) − n+1 n n+1 n+1 n − (4.11) + µ [µ + ...] µ µ n n − n n+1 1 =2pµ 2pµ + ∂ µ + ∂ µ + µ2 2µ µ + µ2 + . n − n+1 ξ n ξ n+1 n − n n+1 n+1 O q Thus, the continuum limit yields

∂ (µ + µ )=2p(µ µ ) (µ µ )2. (4.12) ξ n n+1 n+1 − n − n+1 − n

As the equation contains both discretized, µn, and continuous, ξ, variables, (4.12) is called a differential-difference form of, in this case, the pKdV equation. To complete the limit process, it is necessary to apply a continuum limit a second time to the remaining discrete variable n, using a limit similar to that used in the first step, i.e.,

p with n = τp, for some ﬁxed τ. (4.13) → ∞ The corresponding shift operator form is then n µ (ξ)= υ (ξ,τ) where τ = τ + , (4.14) n 0 p for some ﬁxed initial value τ0. The corresponding Taylor expansion is 1 1 1 µ (ξ)= υ(ξ,τ + )= υ + υ + υ + ... (4.15) n+1 p p τ 2p2 ττ Substituting into (4.12) yields

1 1 1 2 ∂ υ(ξ,τ)+ υ(ξ,τ + ) =2p υ(ξ,τ + ) υ(ξ,τ) υ(ξ,τ + ) υ(ξ,τ) , (4.16) ξ p p − − p − which, based on (4.15), gives

1 1 1 1 1 1 2 ∂ 2υ + υ + υ + ... =2p υ + υ + ... υ + υ + ... . (4.17) ξ p τ 2p2 ττ p τ 2p2 ττ − p τ 2p2 ττ

53 Simplifying,

1 1 1 2 1 2(υ υ )+ ∂ (υ υ )+ υ υ +(υ )2 + =0. (4.18) ξ − τ p τ ξ − τ p2 2 ξττ − 3 3τ τ O p3 Taking the limit, p , in (4.18) would give a linear PDE, → ∞ 2k(ξ+τ) υξ = υτ with solution, υ(ξ,τ)= e− υ(0, 0). (4.19)

To obtain a nonlinear PDE, we return to the original linear solution, (4.4), and apply the ﬁrst continuum limit, q . Hence, → ∞ n n p k 2kξ 1 k/p 2kξ µ (ξ)= − e− = − e− , n p + k 1+ k/p = exp n ln(1 k/p) ln(1 + k/p) 2kξ , − − − (4.20) k 2 k3 2 k5 = exp τp 2 ... 2kξ , − p − 3 p3 − 5 p5 − − 2τ 1 = exp k( 2(τ + ξ)) + k3( ) . − − 3p2 −O p4 3 Then for a non-trivial solution, ekx+k t, deﬁne 2τ x := 2(τ + ξ) and t := . (4.21) − − 3p2 Then the change of variables υ(ξ,τ) w(x,t), changes the partial derivatives as follows, 7→ 1 ∂ = 2∂ and ∂ = 2 ∂ + ∂ . (4.22) ξ − x τ − x 3p2 t Substituting into (4.18) and collecting terms yields,

4 1 w +3w2 + w + =0. (4.23) 3p2 t x xxx O p3 Thus, applying the continuum limit, p , gives → ∞ 2 wt +3wx + wxxx =0, (4.24) the continuous pKdV equation.

54 CHAPTER 5 SYMBOLIC COMPUTATION OF LAX PAIRS

Finding a Lax pair for a given nonlinear equation, whether continuous or discrete, is generally a difficult task. For PDEs the theory of pseudo potentials [83] might lead to a Lax pair, but it only works in certain cases. The most powerful method to find Lax pairs is the dressing method developed by Zakharov and Shabat in 1974 (see, e.g., [84]). Building on the key idea of the dressing method, there exists a straightforward, algorithmic approach to derive a Lax pair [15, 38] for scalar P∆Es that are CAC (see Section 3.5). We will first discuss that algorithmic approach for scalar P∆Es and then the changes necessary to adapt the algorithm to systems of lattice equations.

5.1 Consistency around the cube for scalar P∆Es

Recall from Section 3.5, that the concept of multi-dimensional consistency was introduced independently in [15, 54]. The key idea is to embed the equation consistently into a multi- dimensional lattice by imposing copies of the same equation, albeit with different lattice parameters in different directions. For multi-affine nonlinear P∆Es with the CAC property there is an algorithmic way of deriving a Lax pair. The consistency property does not depend on the actual mappings used to generate the P∆Es on the various faces of the cube. Mappings such as (3.155) which express the symmetries of the P∆Es are merely a tool for generating the needed P∆Es quickly.

Example 5.3.

Consider the lattice modiﬁed KdV (mKdV) equation [31] (also classiﬁed as the H3 equation with δ = 0 as listed in Table 3.2),

p(xx + x x ) q(xx + x x )=0. (5.1) 1 2 12 − 2 1 12

55 This equation is deﬁned on the front face of the cube. To verify CAC, variations of the original P∆E on the left and bottom faces of the cube are generated. Hence, (5.1) is supplemented with two additional equations:

k(xx + x x ) q(xx + x x )=0, (5.2a) 3 2 23 − 2 3 23 p(xx + x x ) k(xx + x x )=0, (5.2b) 1 3 13 − 3 1 13 which yield solutions for x12, x13, and x23:

x(px qx ) x = 1 − 2 , (5.3a) 12 qx px 1 − 2 x(px kx ) x = 1 − 3 , (5.3b) 13 kx px 1 − 3 x(qx kx ) x = 2 − 3 . (5.3c) 23 kx qx 2 − 3 Equations for the remaining faces (i.e., back, right and top) are then generated:

p(x x + x x ) q(x x + x x )=0, (5.4a) 3 13 23 123 − 3 23 13 123 k(x x + x x ) q(x x + x x )=0, (5.4b) 1 13 12 123 − 1 12 13 123 p(x x + x x ) k(x x + x x )=0. (5.4c) 2 12 23 123 − 2 23 12 123

Each of these reference x123 and thus yield three distinct solutions for x123,

x (px qx ) x = 3 13 − 23 , (5.5a) 123 qx px 13 − 23 x (px kx ) x = 2 12 − 23 , (5.5b) 123 kx px 12 − 23 x (qx kx ) x = 1 12 − 13 . (5.5c) 123 kx qx 12 − 13

Remarkably, after substitution of (5.3) into (5.5) one arrives at the same expression for x123, namely,

px x (k2 q2)+ qx x (p2 k2)+ kx x (q2 p2) x = 2 3 − 1 3 − 1 2 − . (5.6) 123 − px (k2 q2)+ qx (p2 k2)+ kx (q2 p2) 1 − 2 − 3 −

56 Thus, (5.1) is consistent around the cube. The consistency is apparent from the following symmetry of the right hand side of (5.6). If we replace the lattice parameters (p,q,k) by (l ,l ,l ) the expression would be invariant under any permutation of the indices 1, 2, 3 . 1 2 3 { }

Additionally, (5.6) does not reference x itself, but depends on x1, x2 and x3. This independence is illustrative of the tetrahedron property, as discussed in Section 3.5.2. Indeed,

through (5.6), the top of a tetrahedron (located at x123) is connected to the base of the

tetrahedron with corners at x1, x2 and x3, as illustrated in Figure 3.4.

5.2 Computation of Lax pairs for scalar P∆Es

Returning to Example 5.3, we show that the CAC property implicitly determines the Lax pair of a P∆E. Indeed, observe that, as a consequence of the multi-aﬃne structure of the

original P∆E, the numerator and denominator of x13 in (5.3b) are linear in x3. In analogy

f with the linearization of Riccati equations, substitute x3 = F into (5.3b), yielding f x(fk Fpx ) x = 1 = − 1 . (5.7) 13 F fp Fkx 1 − 1 Hence,

f = tx(fk Fpx ) and F = t(fp Fkx ), (5.8) 1 − 1 1 − 1 T where t(x,x1; p,k) is a function still to be determined. Deﬁning ψ = f F , the system h i (5.8) can be written in matrix form (2.17a) where L = tLcore and the “core” of the Lax matrix L is given by

kx pxx1 Lcore = − . (5.9) p kx " − 1 # Using (5.3c), the computation of the Lax matrix M proceeds analogously. Indeed,

f x(fk Fqx ) x = 2 = − 2 (5.10) 23 F fq Fkx 2 − 2

57 holds if f = sx(fk Fqx ) and F = s(fq Fkx ) where s(x,x ; q,k) is a common factor 2 − 2 2 − 2 2 to be determined. Thus, we obtain (2.17b) where M = sMcore with

kx qxx2 Mcore = − . (5.11) q kx " − 2 # Note that x can be obtained from x , and hence M from L , by replacing x x 23 13 core core 1 → 2 (or simply, 1 2) and p q. The ﬁnal step is to compute s and t. → → 5.3 Determination of the scalar factors for scalar P∆Es

Speciﬁc values for s and t can be computed using (2.18). Substituting L = tLcore and

M = sMcore yields

st (L ) M ts (M ) L ˙=0. (5.12) 2 core 2 core − 1 core 1 core All elements in the matrix on the left hand side must vanish. Remarkably, this yields a st unique expression for the ratio 2 . ts1 For Example 5.3, using (5.9) and (5.11), eq. (5.12) reduces to

2 2 2 2 2 2 xx1ts1 xx2st2 (k p )qx1 (k q )px2 k(p q )x1x2 0 0 − − −2 2− 2 2 − 2 2 = . px2 qx1 k(p q ) (k p )qx (k q )px 0 0 − " − − − 1 − − 2# " # (5.13) This requires that st x 2 ˙= 1 , (5.14) ts1 x2 which has an inﬁnite family of solutions. Indeed, the left hand side of (5.14) is invariant under the change a a t 1 t, s 2 s, (5.15) → a → a where a(x) is arbitrary. Consistent with the notations in Section 3.5.1, a1 and a2 denote the shifts of a in the 1 and 2 direction, respectively. By inspection, − −

58 1 t = s = , and (5.16a) x 1 1 t = , s = (5.16b) x1 x2 both satisfy (5.14). Note that (5.16a) can be mapped into (5.16b) by taking a =1/x. Avoiding guess work, t and s can be computed using the determinant method. That is, if L and M are n n matrices, then taking the determinant of (5.12) yields core core × n n (st2) det(Lcore)2 det Mcore =(ts1) det(Mcore)1 det Lcore. (5.17)

Thus

st det(M ) detL 2 = n core 1 core , (5.18) ts1 sdet(Lcore)2 detMcore which is satisﬁed by 1 1 t = n , s = n . (5.19) √detLcore √detMcore For Example 5.3, i.e., eq. (5.1), by substituting (5.9) and (5.11) into (5.18), one then obtains 1 1 t = , s = . (5.20) (p2 k2)xx (q2 k2)xx − 1 − 2 The constant factors involvingp p,q and k are irrelevant.p Therefore, (5.20) can be replaced by 1 1 t = , s = . (5.21) √xx1 √xx2 Thus, using the determinant method, a Lax pair for (5.1) is

1 kx pxx 1 kx qxx L = − 1 , M = − 2 . (5.22) √xx1 p kx √xx2 q kx " − 1 # " − 2 # The irrational t and s in (5.21) can be transformed into (5.16a), by taking a = √x, or into

1 (5.16b), by a = √x , both yielding rational Lax pairs.

59 5.4 Consistency around the cube for systems of P∆Es

The previous sections dealt with single (scalar) P∆Es, i.e., equations involving only one field variable (denoted by x). Now we consider systems of P∆Es defined on quadrilaterals involving multiple field variables. Here we will consider examples involving three field variables x,y, and z. Both Figure 1.1 and Figure 3.3 still apply provided we replace the scalar

x by vector x := (x,y,z). Hence, x1 =(x1,y1,z1), x2 =(x2,y2,z2), x12 =(x12,y12,z12), etc. To apply the algorithm in Section 5.2 to systems of P∆Es, it is necessary to maintain consistency for all equations on all six faces of the cube, handle the edge equations in an appropriate way, and ultimately arrive at the same expressions for x123, as well as for y123 and z123.

Example 5.4.

Consider the lattice Schwarzian Boussinesq system [85]:

x y z + z =0, (5.23a) 1 − 1 x y z + z =0, (5.23b) 2 − 2 xy (y y ) y(px y qx y )=0. (5.23c) 12 1 − 2 − 1 2 − 2 1

Eqs. (5.23a) and (5.23b) are deﬁned along a single edge of the square while (5.23c) is deﬁned on the whole square. The edge equations, unlike the face equation, can be shifted in the 1 or 2 directions while still remaining on the square. Then, (5.23) is augmented with − − additional shifted edge equations,

x y z + z =0, (5.24a) 12 2 − 12 2 x y z + z =0, (5.24b) 12 1 − 12 1 obtained from (5.23a) and (5.23b), respectively. Solving for the variables x12 =(x12,y12,z12) referenced in the augmented system (i.e., (5.23) augmented with (5.24)) gives

60 z z x = 2 − 1 , (5.25a) 12 y y 1 − 2 y(px1y2 qx2y1) y12 = − , (5.25b) x(y1 y2) y z y−z z = 1 2 − 2 1 . (5.25c) 12 y y 1 − 2 Continuing as before by generating the variations of (5.23) on the faces of the cube and

solving for the variables with double subscripts yields x13 and x23. Indeed, from the equations

on the bottom face (not shown) one gets x13 with components

z z x = 3 − 1 , (5.26a) 13 y y 1 − 3 y(px1y3 kx3y1) y13 = − , (5.26b) x(y1 y3) y z y−z z = 1 3 − 3 1 , (5.26c) 13 y y 1 − 3 which readily follow from (5.25) by replacing x x , x x , and q k. Or simpler, 2 → 3 12 → 13 → 2 3 and q k. Similarly, the equations on the left face of the cube determine x with → → 23 components

z z x = 2 − 3 , (5.27a) 23 y y 3 − 2 y(kx3y2 qx2y3) y23 = − , (5.27b) x(y3 y2) y z y−z z = 3 2 − 2 3 , (5.27c) 23 y y 3 − 2 easily obtained by a change of labels and parameters, namely, 1 2,p q, 2 3, and → → → q k). Likewise, the equations on the back face (not shown) determine x with components → 123

61 z z x = 23 − 13 , (5.28a) 123 y y 13 − 23 y3(px13y23 qx23y13) y123 = − , (5.28b) x3(y13 y23) y z y −z z = 13 23 − 23 13 , (5.28c) 123 y y 13 − 23 which follow from (5.25) by applying the shift in the third direction, which amounts to “adding” a label 3 to all variables. Similarly, the equations on the right face (suppressed)

yield x123 with components

z z x = 12 − 13 , (5.29a) 123 y y 13 − 12 y1(kx13y12 qx12y13) y123 = − , (5.29b) x1(y13 y12) y z y −z z = 13 23 − 12 13 , (5.29c) 123 y y 13 − 12 which follow from (5.27) by applying a shift in the 1 direction. Finally, the equations on − the top face (suppressed) yield

z z x = 23 − 12 , (5.30a) 123 y y 12 − 23 y2(px12y23 kx23y12) y123 = − , (5.30b) x2(y12 y23) y z y −z z = 12 23 − 23 12 , (5.30c) 123 y y 12 − 23 obtained from (5.26) by a shift in the 2 direction. −

Using (5.25)-(5.27) to evaluate the expressions (5.28)-(5.30) yields the same x123 with

x(x1 x2) y1(z2 z3)+ y2(z3 z1)+ y3(z1 z2) x123 = − − − − , (5.31a) (z z ) px (y y )+ qx (y y )+ kx (y y ) 1 − 2 1 3 − 2 2 1 − 3 3 2 − 1 q(z2 z1)( kx3y1 px1y3)+ k(z3 z1)(px1y2 qx2y1) y123 = − − − − , (5.31b) x px (y y )+ qx (y y )+ kx (y y ) 1 1 3 − 2 2 1 − 3 3 2 − 1 px (y z y z )+ qx (y z y z )+ kx (y z y z ) z = 1 3 2 − 2 3 2 1 3 − 3 1 3 2 1 − 1 2 . (5.31c) 123 px (y y )+ qx (y y )+ kx (y y ) 1 3 − 2 2 1 − 3 3 2 − 1

62 Thus, (5.23) is multi-dimensionally consistent around the cube, i.e., the systems of P∆Es is consistent around the cube with respect to each component of x, i.e., x, y and z.

The expressions for x123 and y123 can be written in more symmetric form by eliminating z1,z2, and z3. To do so, we use the edge equations

x y z + z =0, (5.32a) 3 − 3 x y z + z =0, (5.32b) 2 − 2 deﬁned on the left face of the cube. Subtracting (5.23a) from (5.23b) and (5.32a) from (5.32b) yields z z z z z z 2 − 1 = 3 − 2 = 3 − 1 = y. (5.33) x x x x x x 2 − 1 3 − 2 3 − 1 Using the above ratios, (5.31a) and (5.31b) can be replaced by

x y1(x2 x3)+ y2(x3 x1)+ y3(x1 x2) x = − − − , (5.34a) 123 px (y y )+ qx (y y )+ kx (y y ) 1 3 − 2 2 1 − 3 3 2 − 1

y kqy1(x2 x3)+ kpy2(x3 x1)+ pqy3(x1 x2) y = − − − . (5.34b) 123 px (y y )+ qx (y y )+ kx (y y ) 1 3 − 2 2 1 − 3 3 2 − 1 Before continuing with the calculations of a Lax pair, it is worth noting that (5.23) does not satisfy the tetrahedron property because x explicitly appears in the right hand side of (5.31a). The impact of not having the tetrahedron property remains unclear but does not aﬀect the computation of a Lax pair.

5.5 Computation of a Lax pair for systems of P∆Es

Both the numerators and denominators of the components of x13 and x23 (in (5.26) and

(5.27), respectively), are aﬃne linear in the components of x. Due to their linearity in x3,

y3 and z3, substitution of fractional expressions for x3, y3 and z3 will allow one to compute Lax matrices. In contrast to the scalar case, the computations are more subtle because the

edge equations on the left face of the cube introduce constraints between x3 and z3.

63 Continuing with Example 5.4, solving (5.32a) for x3 yields z z x = 3 − . (5.35) 3 y Therefore, setting f g z = and y = (5.36a) 3 F 3 G determines z z f Fz x = 3 − = − . (5.36b) 3 y Fy Substituting (5.36) into (5.26) then yields

G(Fz f) x = 1 − , (5.37a) 13 F (g Gy ) − 1 Gfky Fgx y FGky z y = 1 − 1 − 1 , (5.37b) 13 Fx(g Gy ) − 1 Fgz Gfy z = 1 − 1 , (5.37c) 13 F (g Gy ) − 1 which are not yet linear in f, g, F and G. Additional constraints between f, g, F and G will achieve this goal. Indeed, setting G = F simpliﬁes (5.37) into f Fz x = − 1 , (5.38a) 13 Fy g 1 − gpx y fky + Fky z y = 1 − 1 1 , (5.38b) 13 x(Fy g) 1 − fy gz z = 1 − 1 . (5.38c) 13 Fy g 1 − Simultaneously, (5.36) reduces to f g z = , and y = , (5.39a) 3 F 3 F f Fz x = − , (5.39b) 3 Fy

64 f1 whose shifts in the 1 direction must be compatible with (5.38). Equating z13 = to (5.38c) − F1 requires that

f = t (fy gz ) (5.40) 1 1 − 1 and

F = t (Fy g). (5.41) 1 1 − Next, equating y = g1 with (5.38b) gives 13 F1 1 g = t (gpx y fky + Fky z). (5.42) 1 x 1 − 1 1 Finally, one has to verify that the 1-shift of (5.39b),

f1 F1z1 x13 = − , (5.43) F1y1

matches (5.38a). That is indeed the case. After substitution of f1 and F1 into (5.43) t(fy gz ) t(Fy g)z f Fz x = 1 − 1 − 1 − 1 = − 1 . (5.44) 13 t(Fy g)y Fy g 1 − 1 1 − T Deﬁning ψ = gf F , eqs. (5.40)-(5.42) can be written in matrix form yielding (2.17a) with h i

px1y ky1 ky1z x − x x L = t z y 0 , (5.45) − 1 1  1 0 y  − 1    where t(x, x1; p,k). Similarly, from (5.27) one derives

qx2y ky2 ky2z x − x x M = s z y 0 , (5.46) − 2 2  1 0 y  − 2    which can also be obtained from (5.45) by applying the replacement rules 1 2 and p q. → →

65 5.6 Determination of the scalar factors for systems of P∆Es

As discussed in Section 5.3, speciﬁc values for s and t may be computed algorithmically using (5.19). For Example 5.4, this yields 1 1 t = , s = . (5.47) 2 2 3 (k p)y (z z1) 3 (k q)y (z z2) − 1 − − 2 − x x Cancelling trivial factors, a Laxq pair for (5.23) is thusq given by

px1y ky1 ky1z x x − x x 3 L = 2  z1 y1 0  , (5.48a) y (z z1) − r 1 − 1 0 y  − 1    qx2y ky2 ky2z x x − x x 3 M = 2  z2 y2 0  . (5.48b) y (z z2) − r 2 − 1 0 y  − 2    Unfortunately, these matrices have irrational functional factors. Using (2.18) we ﬁnd the following equation for the scalar factors st y 2 ˙= 1 . (5.49) ts1 y2 Once can easily verify that (5.49) is satisﬁed by 1 1 1 t = s = and t = , s = , (5.50) y y1 y2 which both yield rational Lax pairs. The factors t,s in (5.50) are related to those in (5.47). Using (5.23a), t in (5.47) can be written as

x t = 3 . (5.51) (p k)y2yx r − 1 1 3 After applying (5.15) with a = x/y, one can simplify the cube root to ﬁnd t =1/y1, where the trivial factor 1/√3 p k hasp been canceled. A further application of (5.15) with a = y − then yields t =1/y. The connections between the choices for s are similar. An alternate form of a Lax pair is possible. Had the original constraint given by (5.32a) been expressed as

z3 = x3y + z, (5.52)

66 the substitutions would become

f˜ g˜ x3 = , and y3 = , (5.53a) F˜ F˜ fy˜ + Fz˜ z3 = . (5.53b) F˜ T With φ = f˜ g˜ F˜ , would then be given by L h i px1y kyy1 0 x − x = t 0 y z z . (5.54) L  − 1 1 0 y  − 1    Note that the matrices (5.45) and (5.54) are gauge equivalent as deﬁned in (2.19) with

10 0 = 0 1/y z/y . (5.55) G  −  00 1    

67 Table 5.1: Edge Constrained Systems of P∆Es

System

Name Equation Edge ψ Matrix L Scalar Alternate Ref. Constraint Ratio t values

y xx1 0 1 z1 xx1 + y = 0, − 2 2 − F t z(xx1 y) x y1 + x(p k) x z , Lattice z2 xx2 + y = 0, y + z3   st2 x1 − x = ψ = − − − − = [86] (x2 x1)(z xx12 + y12) 3 a  g  yy1 xx1 y1 Boussinesq − − x  − −  ts1 x2 p + q =0. h  1  −   with t = .   x

x1 1 0 F − t  y1 0 1 , st2 z3 = xx3 y ψb = f  − =1 − x1z xy1 + p k z x ts1 g  − − −        with t = 1.

y1z x1 + x =0, z1 0 1 − − Modiﬁed/ y2z x2 + x =0, F t 0 z1 x1 , − x3 x  3 − 3 3  st2 z1 1 z12(z1 z2) k xz1 k z1 p y1z = Schwarzian − y3 = − ψa = f  t = [20, 87] z z − y − y − y  ts1 z2 z1 Boussinesq 3 3 h   y1z2p y2z1q =0.   1 − y −   with t = . z

z1 0 1 − F t x1 x z 0 ,  − −  st2 z1 1 z 3 z 3 = x3 = y3z + x ψb =  g  0 k z1 p y1 t =  y − y  ts1 z2 z1 h  1    with t = .   z

x1z y1 x =0, − − zz1 0 z x2z y2 x =0, − − y − F t xz1 z1 x1z , x + y3   st2 z1 1 Hietarinta z12 b0 kx−px1z−zz1(b0x+y) z(−b0x+y) − x − x3 = ψa =  g  k = t = [20] A-2 z  x x x  ts1 z2 z1 1 px1 qx2 h  1  − =0.   with t = . − x z1 z2   z −

z1 0 1 F − t x 1 0 , st2 y = zx x ψ =  1  =1 3 3 b f  px1+−z1(b0x+y) k b0x+y − ts1 h − x x x        with t = 1.

68 Table 5.1: Continued.

System

Name Equation Edge ψ Matrix L Scalar Alternate Ref. Constraint Ratio t values

x1 1 0 − t z1 0 1 , xx1 y1 z =0 −  − − ℓ31 a0 b0x y x + b0 xx2 y2 z =0 F  − −  − − st2 Hietarinta z12 a0 + b0(x x12)+ y with t = 1 and where  y3 = xx3 z ψa = f  =1 [20] B-2 − − − ts1 p q h xx12 − =0   ℓ31 =(k p) x1(a0 b0x y) − − x1 x2   − − − − − z1(x + b0) −

xx1 z 1 0 − − t zz1 z1 xx1 ,  −  F ℓ31 (a0 b0x y) z(x + b0) z + y3  − − − −  st2 x1 1 x3 = ψb =  g   1  = t = x with t = and where ts1 x2 x1 h x     ℓ31 = x(k p)+(xx1 z)(a0 b0x y) − − − − − + xz1(x + b0)

zy1 x1 + x =0 − z1 0 1 zy2 x2 + x =0 − − 1 F t x1 x z 0 ,   st2 z1 1 z12 d2x + d1 z1(d−2x+d1) kzz− 1 d2x+d1−pzy1 Hietarinta − y − x3 = zy3 + x ψa =  g  = t = [20] − y y y  ts1 z2 z1 C-3 h   z py1z2 qy2z1   1 − =0   with t = y z1 z2 z −

z1 0 1 − F t 0 z1 x1 , x3 x   st2 z1 1 z1(d2x+d1+kx) −kx1 d2x+d1−pzy1 y3 = − ψb = f  = t = z − y y y  ts1 z2 z1 h  1    with t =   z

69 Table 5.1: Continued.

System

Name Equation Edge ψ Matrix L Scalar Alternate Ref. Constraint Ratio t values

y1z x1 + x =0 − z1 0 1 y2z x2 + x =0 − −xx d F t x1 x z 0 , Hietarinta 12 1  − −  st2 z1 1 z12 − x = x + y z ψ = g z1 2 zz1 = t = [20] − y − 3 3 a   y x + d1 y k x ℓ33 C-4 − −  ts1 z2 z1 h  1 1  z py1z2 qy2z1   with t = , and where ℓ = xx pzy + d . − =0   33 1 1 1 y z1 z2 z y − − z1 0 1 − F t 0 z1 x1 , x3 x  −  st2 z1 1 y = − ψ = f z1 z1 = t = 3 b   y d1 + kx y k x ℓ33 z − −  ts1 z2 z1 h  1 1    with t = , and whereℓ33 = xx1 pzy1 + d1   z y −

Table 5.2: Additional Systems of P∆Es

System

Name Equation ψ Matrix L Scalar Alternate Ref. Ratio t values

z1 0 1 x2z1 x1z2 − t 0 z1 x1 , x12 − =0 F  −  − z1 z2 st2 z1 1 Hietarinta kzz1 zz1 z(x1 p) − f   − −  = t = [20] C-2.1 z(pz2 qz1)   ts1 z2 z1 z12 + zx12 − =0 h − z1 z2   1 −   with t = . z

z1 0 1 x2z1 x1z2 − x12 − =0 F t  0 z1 x1 , − z1 z2 z z− z st2 z1 1 Hietarinta (d0z1) (kz1) (d0 + px1) − f   x x x  = t = [20] C-2.2 z z px1z2 qx2z1)   ts1 z2 z1 z12 + d0 − =0 h x − x z1 z2   1 −   with t = . z

70 Table 5.2: Continued.

System

Name Equation ψ Matrix L Scalar Alternate Ref. Ratio t values

Modiﬁed F py1 0 k x12(py1 qy2) y(px2 qx1)=0, − 1 st2 y1 1 Boussinesq − − − f  t  kx1y py 0  with t = = t = [88] xy12(py1 qy2) y(px1y2 qx2y1)=0. − y ts1 y2 y1 (2-component) − − − g 0 kyy1 px1y    − x x     

p k x 1 0 y12(p q + x2 x1) 1 − − − − 1 2 t ℓ21 p + k z (k + k + 1) (p 1)y2 +(q 1)y1 =0  y  − − − − F y1(1 k) 0 p 1 y1y2(p q z2 + z1)  − −  st Toda-Modiﬁed − −   2 f  with t = 1, and where =1 [89] Boussinesq (p 1)yy2 +(q 1)yy1 =0 ts1 − − − g y1 2 y(p + q z x12)(p q + x2 x1)   ℓ21 =(p + k z)(p k x1) (p + p + 1) − − − −   − − − − 2 2 y (p + p + 1)y1 +(q + q + 1)y2 =0 −

y1 y2 y((x1 x2)y + p q)=0 F k p yx1 y st2 Lattice NLS − − − − t − − with t =1 =1 [90, 91] x1 x2 + x12((x1 x2)y + p q =0 f x1 1 ts1 − − − " # " − #

2 2 0 0 tx t(p k xy1) f − − 2 2 0 0 t ty1 lpKdV (x x12)(y1 y2) p + q =0 F  −  − − −   Ty T (p2 k2 x y)0 0 [90, 91] (2-component) 2 2 1 (y y12)(x1 x2) p + q =0 g  − −  − − −    T Tx1 0 0  G  −        with T = t =1

71 CHAPTER 6 GAUGE EQUIVALENCES

In [20], Hietarinta derived a family of Boussinesq-type P∆Es. Shortly thereafter, in [32], Bridgman et al. derived the corresponding Lax pairs using [17]. Subsequently, Zhang et al. [92] showed that each of the lattice systems presented in [20] can be further generalized based on a direct linearization scheme [89] in connection with a more general dispersion law. The systems considered in [20] are then shown to be special cases. In fact, they are connected to the more general cases through point transformations. As a by-product of the direct linearization method, Zhang et al. [92] obtained the Lax pairs of each of these extended lattice Boussinesq-type systems. No doubt, they are all valid Lax pairs but some of the matrices have larger than needed sizes. Using the algorithmic CAC approach, Bridgman et al. [33] derived Lax pairs of minimal matrix sizes13 for these systems and unraveled the connections with the Lax matrices presented in [92]. In addition to presenting the work of [33] in this dissertation, we identify the gauge transformations between the Lax pair of similar dimension presented in [92] and discuss the gauge-like relationships that exist between the Lax pairs of diﬀering dimensions. We call the transformation gauge- like because the transformation matrices are no longer square. In the process, we also discuss the proper treatment of the edge equations in the derivation of the Lax pair and the general form of the dispersion relation used in [92].

6.1 Gauge Equivalence of Lax Pairs for PDEs and P∆Es

As discussed in Chapter 2, analogous with the deﬁnition of Lax pairs (in matrix form) for PDEs, a Lax pair for a system of P∆Es is a pair of matrices, (L,M), such that the compatibility of the linear system,

ψ1 = Lψ and ψ2 = Mψ, (6.1) 13 The resulting matrix dimension is based on the substitution process described in Chapter 5.

72 for an auxiliary vector function ψ, requires that the P∆E is satisﬁed. Also, in analogy to (2.9), the Lax pair (L,M) must satisfy

L M M L ˙=0, (6.2) 2 − 1 where ˙=denotes that the equation holds for solutions of the P∆E. (6.2) is called the Lax equation (or zero-curvature condition). As in the continuous case, a Lax pair for a given system of P∆Es is not unique. Here again, there is an inﬁnite number of Lax matrix pairs, all equivalent to each other under gauge transformations [17]. Speciﬁcally, if (L,M) is a Lax pair then so is (L,ˆ Mˆ ) where

1 1 Lˆ = L − and Mˆ = M − , (6.3) G1 G G2 G for any arbitrary invertible matrix . Gauge transformation (6.3) comes from setting ψˆ = ψ G G and requiring that ψˆ1 = Lˆψˆ and ψˆ2 = Mˆ ψ.ˆ

6.2 Derivation of Gauge Equivalent Lax Pairs

Consider the generalized Hietarinta A-2 system (henceforth, the A-2 system) as given in [92],

zx y x =0, zx y x =0, and (6.4a) 1 − 1 − 2 − 2 − G( p, a)x G( q, a)x y xz + b x + − − 1 − − − 2 =0, (6.4b) − 12 0 z z 2 − 1 with the dispersion relation G(ω,κ) deﬁned as

G(ω,κ) := ω3 κ3 + α (ω2 κ2)+ α (ω κ). (6.5) − 2 − 1 −

The term with coeﬃcient b0 could be removed by a simple transformation [20]. We will keep it to cover the most general case. For the special case a = α = α =0, one gets G( p, a)= p3 and G( q, a)= q3. 1 2 − − − − − − Then (6.4) reduces to Hietarinta’s original A-2 system in [64, p. 95]. In [17] and [20], p3 and q3 are identiﬁed with p and q, respectively. They are called the lattice parameters.

73 Note that system (6.4) is a multi-component system with field variables x,y, and z and their shifts. Eq. (6.4b) is defined on the quadrilateral while the equations in (6.4a) are each restricted to a single edge of the quadrilateral depicted in Figure 1.1 where x denotes (x,y,z). The way in which the edge equations are handled in the process of deriving a Lax pair, will yield different, yet gauge-equivalent Lax pairs. As discussed in Chapter 3, the key idea of multi-dimensional consistency is to (i) extend the planar quadrilateral (square) to a cube by artificially introducing a third direction (with lattice parameter k) as shown in Figure 3.3, (ii) impose copies of the same system, albeit with different lattice parameters, on the different faces and edges of the cube, and (iii) view the cube as a three-dimensional commutative diagram for x123,y123, and z123. Consistency around the cube of the P∆E system requires that one can uniquely determine

x123 =(x123,y123,z123) and that all expressions coincide, no matter which path along the cube is followed. As discussed in [31], this three-dimensional consistency establishes integrability for it allows one to algorithmically compute a Lax pair. In [17], a simpliﬁed version of (6.4) (with b = 0 and G( p, a) and G( q, a) replaced 0 − − − − by p and q, respectively) was shown to be 3D consistent. The way to verify the multi- − − dimensional consistency of (6.4) is similar. For (6.4), after some algebra, the components of

x13 are

x x x z x z x = 1 − 3 , y = 1 3 − 3 1 , and (6.6a) 13 z z 13 z z 1 − 3 1 − 3 (y + b x)(z z )+ G( k, a)x G( p, a)x z = 0 1 − 3 − − 3 − − − 1 , (6.6b) 13 x(z z ) 1 − 3

x1z2 x2z1 where we have used that the edge equations are compatible if and only if y = − , 12 z1 z2 −

which yields the expressions for y13 and, consequently, x13 in (6.6a). The expression for z13 in (6.6b) readily follows from (6.4b) after replacing indices 2 by 3 and q by k. Interestingly, for the A-2 system (6.4), veriﬁcation of 3D consistency and, consequently, the derivation of Lax pairs does not require the explicit form of G(ω,κ) in (6.5). Neither

74 does it require that

G( p, b) G( p, a)= G( a, b), (6.7) − − − − − − − holds. This condition was given in [92, p. 231]. Condition (6.7) is of the form

G(a, c)+ G(c,b)= G(a, b). (6.8)

Its general solution is G(a, b) = f(a) f(b) for an arbitrary function f. The proof [93] − proceeds as follows: Setting c = b = a yields G(a, a)=0. Next, setting b = a yields G(c,b) = G(b,c), expressing skew-symmetry. Setting c = c , where c is any convenient − 0 0 value, yields G(a, b)= G(a, c ) G(b,c ). Deﬁning f(a)= G(a, c ) establishes the result. 0 − 0 0

In particular, (6.7) is satisﬁed for the following monic (i.e., leading coeﬃcient αN = 1)

with coeﬃcients αj,

N N N 1 N 1 N 2 N 2 Ggeneral(ω,κ) := ω κ +αN 1(ω − κ − )+αN 2(ω − κ − )+...+α1(ω κ), (6.9) − − − − − −

for any integer N 2. Expression (6.9) conﬁrms the admissibility of the general (parameter- ≥ ized) dispersion law discussed in [92]. However, as shown in Section 2 of [92], the derivation of (6.4) by direct linearization required that N =3. We now derive the Lax pair for (6.4), ignoring the explicit expressions for G( p, a) and − − G( q, a). Analogous with the linearization of Riccati equations, we introduce projective − − variables f,g,h,F,G, and H by f g h x = , y = , and z = . (6.10) 3 F 3 G 3 H After substitution of (6.10) into (6.6) we check if the numerators and denominators of the resulting expressions are linear in the projective variables. If not, we reduce the number of projective variables till we achieve linearity. For example,

H(x F f) x = 1 − , (6.11) 13 F (z H h) 1 − will be linear in top and bottom if H = F. Also, the expressions of y13 and z13 are then both linear in top and bottom. Thus far, G is undetermined. However, using the (derived) edge

75 equation,

zx y x =0, (6.12) 3 − 3 − F (xG+g) yields f = zG , which becomes linear in top and bottom if G = F. Then, xF + g f = (6.13) z and xF + g g h x = , y = , and z = , (6.14) 3 zF 3 F 3 F where F,g, and h are independent (and remain undetermined). Eqs. (6.6), then become

f x F + g (x zx )F + g g z (xF + g) zx h x = 1 = 1 1 1 = − 1 , y = 1 = 1 − 1 , and (6.15a) 13 F z F z(h z F ) 13 F z(h z F ) 1 1 1 − 1 1 − 1 h (y + b x)z(h z F ) G( k, a)(xF + g)+ G( p, a)zx F z = 1 = 0 − 1 − − − − − 1 . (6.15b) 13 F xz(h z F ) 1 − 1

The equations above for y13 and z13 can be split by setting

F = t ( zz F + zh), g = t (xz F + z g zx h), (6.16a) 1 − 1 1 1 1 − 1 t h = G( p, a)zx G( k, a)x (y + b x)zz F 1 x − − 1 − − − − 0 1 G( k, a)g +(y + b x)zh , (6.16b) − − − 0

where t(x,y,z,x1,y1,z1; p,k) is a scalar function still to be determined. One can readily

verify that the equations for x13 in (6.15) are then identically satisﬁed. T T If we deﬁne ψa := F gh , then (ψa)1 = F1 g1 h1 , where T stands for transpose, we h i h i can write (6.16) in matrix form (ψa)1 = Laψa, with

zz 0 z − 1   La = tLa,core := t xz z zx , (6.17)  1 1 − 1       G( k, a) (y+b0x)z   ℓ31 − −   − x x  1   where ℓ = G( p, a)zx G( k, a)x (y + b x)zz . 31 x − − 1 − − − − 0 1

76 The partner matrix Ma of the Lax pair is then,

zz 0 z − 2   Ma = sMa,core := s xz z zx , (6.18)  2 2 − 2       G( k, a) (y+b0x)z   m31 − −   − x x  1   where m = G( q, a)zx G( k, a)x (y + b x)zz . The matrix M follows 31 x − − 2 − − − − 0 2 a,core from La,core by replacing all indices 1 by 2 and p by q (see, e.g., [17] for details). In subsequent examples, the partner matrices (M) will no longer be shown.

Using the same terminology as in [17], La,core and Ma,core are the “core” of the Lax matrices La and Ma, respectively. The label “a” on ψa,La, and Ma is added to indicate a ﬁrst choice for deﬁning ψ (up to trivial permutations of the components). In what follows, alternative choices will be labeled with “b” and “c” (i.e., ψb,ψc,Lb,Lc, etc.).

The functions t(x, x1; p,k) and s(x, x2; q,k) can be computed algorithmically as shown in [17] or by using the Lax equation (6.2) directly, as follows,

(tL ) (sM ) (sM ) (tL )=(st )(L ) M (ts )(M ) L ˙=0, (6.19) core 2 core − core 1 core 2 core 2 core − 1 core 1 core

which implies that

st2 (Lcore)2 Mcore ˙=(Mcore)1 Lcore. (6.20) ts1

After replacing Lcore and Mcore by La,core and Ma,core from (6.17) and (6.18), respectively, in (6.20), one gets st z 2 ˙= 1 , (6.21) ts1 z2 which has an inﬁnite family of solutions. Indeed, the left hand side of (6.21) is invariant under the change i i t 1 t, s 2 s, (6.22) → i → i

77 where i(x,y,z) is an arbitrary function and i and i denote the shifts of i in the 1 and 1 2 − 2 direction, respectively. One can readily verify that (6.21) is satisﬁed by, for example, − 1 s = t = . (6.23) z T Then, for ψa = F gh , the Lax matrix La in (6.17) becomes h i zz 0 z − 1 1   La = xz z zx . (6.24) z  1 1 − 1       G( k, a) (y+b0x)z   ℓ31 − −   − x x  With regard to (6.3), it is clear that Lax pair are not unique. In particular, since the edge constraint (6.12) is linear in both x3 and y3, we can construct a gauge-equivalent Lax pair

by treating the edge constraint in a diﬀerent way. Indeed, solving (6.12) for y3 (instead of

x3) yields

y = zx x. (6.25) 3 3 − Thus, f g zf xF h x = , y = = − , and z = , (6.26) 3 F 3 G F 3 F yield G = F and

g = zf xF. (6.27) − Then,

f x F z f x h x = − 1 , y = 1 − 1 , and (6.28a) 13 h z F 13 h z F − 1 − 1 (y + b x)(h z F )+ G( p, a)x F G( k, a)f z = 0 − 1 − − 1 − − − . (6.28b) 13 x(h z F ) − 1

78 T Thus, for ψb := Ffh , upon computation of the corresponding scalar functions resulting in s = t = 1, oneh gets i

z 0 1 − 1   Lb = x 1 0 , (6.29)  − 1       G( p, a)x1 (y+b0x)z1 G( k, a) y+b x   − − − − − 0   x − x x    where again Mb follows from Lb by changing indices 1 into 2 and replacing p by q. Surprisingly, (6.24) and (6.29) are not the only Lax pairs that can be calculated based on the 3D consistency of the system. We now show how the Lax pair in [92], involving 4 4 × matrices, was computed. If one simply substitutes f g h x = , y = , and z = (6.30) 3 F 3 F 3 F into (6.6) one gets,

f x F z f x h x = − 1 , y = 1 − 1 , and (6.31a) 13 h z F 13 h z F − 1 − 1 (y + b x)(h z F )+ G( p, a)x F G( k, a)f z = 0 − 1 − − 1 − − − . (6.31b) 13 x(h z F ) − 1 The numerators and denominators of all three expressions in (6.31a) and (6.31b) are linear in F,f,g, and h, although g does not appear explicitly. Once can thus set up a 4 4 Lax matrix T × with a zero column. Indeed, for ψc := Ffgh , upon calculation of the corresponding scalar functions yielding s = t = 1, one obtainsh i

z 0 01 − 1   x 1 00  1   −  Lc =   . (6.32)    0 z1 0 x1   −     G G   ( p, a)x1 (y+b0x)z1 ( k, a) y+b0x   − − − − − 0   x − x x   

79 As noted in [92], the zero column in Lc can be removed. Removing the third row and third column in Lc then results in (6.29). Thus, the component g in ψc is redundant. Indeed, g is the numerator of y3, which did not appear in (6.6).

6.3 Generalized Hietarinta Systems

Briefly, the approach discussed in the previous section consists of imposing the edge constraint either at different steps in the Lax pair derivation process or by imposing different representations of the constraint on the derivation process. We can now apply this approach to the generalized Hietarinta family of equations and investigate the gauge equivalences of the resulting Lax pairs.

6.3.1 A-2 System

We now will present the explicit gauge equivalences of the three Lax pairs computed in Section 6.2. Computation of the gauge matrix, , such that G 1 L = L − , (6.33) b G1 aG is straightforward if we consider the implications of (6.3). Indeed, multiplying (6.33) by ψb, and using (6.1) yields

1 (ψ ) = L ψ =( L − )ψ . (6.34a) b 1 b b G1 aG b

1 Hence, if we set ψ = − ψ , we obtain (ψ ) = L ψ = (ψ ) = ( ψ ) , conﬁrming a G b b 1 G1 a a G1 a 1 G a 1 that ψ = ψ . Using (6.13), we get b G a 1 0 0 1 00 x 1 1 = 0 and − = x z 0 , (6.35) G  z z  G −  0 0 1 0 01     because    

1 0 0 F F F ψ = x 1 0 g = xF +g = f = ψ . (6.36) G a  z z     z    b 0 0 1 h h h                

80 Thus, (6.24) and (6.29) are gauge equivalent, as in (6.33), for in (6.35). In essence, G G represents the edge constraint in the system of P∆Es.

We now show the connection between Lc and La given in (6.24). Using (6.13), the linearity of f in F and g allows one to express ψc in terms of ψa in a unique way:

F F 1 0 0 xF +g x 1 F f z z z 0 ψc =   =   =   g := ψa, (6.37) g g 0 1 0   H       h h  h  0 0 1                 deﬁning the 4 3 matrix which expresses the edge constraint in the system of P∆Es. × H Since rank = 3, matrix has a 3-parameter family of left inverses, H H 1+ αx αz α 0 1 − − = (β 1)x (1 β)z β 0 , (6.38) HLeft,all  − −  γx γz γ 1  −    where α,β, and γ are free parameters (which could depend on x).

1 The next step is to take a speciﬁc member (henceforth denoted by − ) of the family HLeft so that

1 L = L − . (6.39) c H1 aHLeft A straightforward matrix multiplication shows that this requires that α = β = γ =0. Hence,

1 000 1 − = x z 0 0 . (6.40) HLeft −  0 001     Instead of (6.33) we now have the transformation (6.39) which can readily be veriﬁed. Indeed, using (6.39), ψ = ψ , and (6.1), yields c H a 1 (ψc)1 = Lcψc = 1La Left− ψa = 1Laψa = 1 (ψa)1 = ψa , (6.41) H H H H H H 1 conﬁrming (6.37). Obviously, plays the role of the gauge matrix but since is not square H H we call it a gauge-like matrix. Likewise, (6.39) is called a gauge-like transformation.

81 The connection between Lc and Lb is simpler. As stated earlier, removing the third row and third column of Lc in (6.32) gives Lb in (6.29). Formally,

L = L T (6.42) b B cB with matrix

1000 = 0100 . (6.43) B   0001     1 Substituting L from (6.39) into (6.42), and using the expressions for and − from c H1 HLeft (6.37) and (6.40), yields

1 T 1 L = L − = L − , (6.44) b BH1 aHLeftB G1 aG

1 for and − in (6.35). Since = , we can factor as G G B1 B G = . (6.45) G BH Thus,

1 1 1 1 1 T − =( )− = − − = − , (6.46) G BH HLeftBRight HLeftB provided we take T as the preferred right inverse of (among inﬁnitely many choices). Eq. B B (6.45) shows that comes from after removing the third row. Likewise, (6.46) shows that G H 1 1 − comes from − after removing the (redundant) third column. G HLeft

Alternate Way. If Lc is known then La can be computed from (6.39) as

1 La = Left− Lc . (6.47) H 1 H Alternatively, it is possible to express La in terms of Lc using right inverses. To do so, start with

F F F 1 000 f ψa = g = xF + zf = x z 0 0   := ψc, (6.48)   −  −  g H h h 0 001         h         e  

82 defining a 3 4 matrix . Note that the matrix is not unique. From (6.37) and (6.38), × H H 1 we see that ψ = ψ with = − . Thus, there is a three parameters family of a Hall c e Hall HLeft,all e choices for . We will continue with the specific choice of defined in (6.48) which has a H e e H 3 parameter family of right inverses, − e e 1 0 0 x 1 1 z z 0 − =   , (6.49) HRight,all α β γ   0 0 1 e     where α,β, and γ are free parameters. However, the inverse transformation,

F 1 0 0 x 1 F f z z 0 1 ψc =   =   g := − ψa, (6.50) g 0 1 0   HRight     h h 0 0 1         e     ﬁxes the right inverse to be used, corresponding to α = γ = 0 and β = 1 in (6.49). Instead of (6.39), the gauge-like transformation then becomes

1 L = L − , (6.51) a H1 cHRight which can be readily veriﬁed as follows. Usinge e (6.1) repeatedly, together with (6.51) and (6.50), yields

1 (ψa)1 = Laψa = 1Lc Right− ψa = 1Lcψc = 1 (ψc)1 = ψc , (6.52) H H H H H 1 conﬁrming (6.48). e e e e e Thus, for the A-2 system (and also for the B-2 system discussed in the next section)

there are two ways of expressing gauge-like transformations between La and Lc, namely (6.39) and (6.51). Of course, and are related. Indeed, comparing (6.47) with (6.51) H H 1 1 yields = − and = − . H HRight H HLeft e As we wille show ine Sections 6.3.3 and 6.3.4, for the C-3 and C-4 lattices the approach with the left inverses does no longer work.

83 The transformation between Lc and Lb given in (6.29) is easy to derive. Using (6.27),

F F 1 00 F f f 0 10 ψc =   =   =   f := ψb, (6.53) g xF + zf x z 0   J   −  −  h h  h   0 01                 one defines which expresses the edge constraint in the system of P∆Es. J Since rank = 3, matrix has infinitely many left inverses (not shown). Next, J J F F 1000 f 1 ψb = f = 0100   := − ψc, (6.54)     g JLeft h 0001       h         defines a left inverse to be used. Note that the selected left inverse in (6.54) is , which B

simply removes the component g in ψc. Finally, the gauge-like transformation becomes

1 L = L − , (6.55) c J1 bJLeft

1 where the multiplication of L by − on the right simply inserts a column with zeros b JLeft

between the second and third columns of Lb, leaving the other columns unchanged. Note

that the third column of Lc in (6.32) had all zeros as well. If the alternate way were used, one would get ψ = ψ and b J c 1 L = L − , e (6.56) b J1 cJRight with e e

1 00 1 000 x 1 1 0 10 = 0 0 and − =   . (6.57) J  z z  JRight x z 0 0 001 −     0 01 e   e     This shows that there are multiple ways to extract Lb from Lc, (6.42) being the simplest. A summary of the Lax pairs for the A-2 system and how they are connected is given in Table 6.1.

84 6.3.2 B-2 System

Revisiting the additional systems presented in [92], we have found similar results. Con- sider the generalized Hietarinta B-2 system (henceforth, the B-2 system),

xx z y =0, xx z y =0, and (6.58a) 1 − 1 − 2 − 2 − G( p, q) y + α + z + α (x x) xx + − − =0, (6.58b) 12 1 2 12 − − 12 x x 2 − 1 with G(ω,κ) deﬁned in (6.5). For comparison with [92], we stay true to α1 and α2 in (6.58b) which are the constants also used in (6.5). Note that the results below are obtained without

using (6.5) explicitly and therefore hold for arbitrary α1 and α2.

In any case, the constant α1 is irrelevant for it can be eliminated [20]. For the special case G( p, q)= p3 q3 and α = b (any constant), system (6.58) reduces to Hietarinta’s − − − 2 0 B-2 system in [64, p. 95]. In [17] and [20], p3 and q3 are identiﬁed with p and q, respectively. We now derive Lax pairs for (6.58), ignoring the explicit expression for G( p, q). Similar − − to the A-2 system, one can derive gauge-equivalent 3 3 Lax matrices by introducing the × edge constraint,

xx z y =0, (6.59) 3 − 3 − to eliminate unknowns in (6.30). Again, there are two choices: one can either eliminate f T T and work with ψa := F gh , or eliminate h and work with ψb := Ffg . T Working with (6.30)h andiψ := Ffgh , yields the 4 4 matrixh L ini Table 6.2 with c × c a redundant column of zeros [92]. h i

The computation of the Lax matrices La and Lb as well as their connection proceeds as in Section 6.3.1. The results are presented in Table 6.2. Based on similar arguments as for the A-2 system, we conclude that comes from G H 1 1 after removing the fourth row and − comes from − after removing the (redundant) G HLeft fourth column.

85 Note, however, that for the B-2 system, the condition

G( p, k)+ G( k, q)= G( p, q), (6.60) − − − − − − which is equivalent with (6.7), must hold for 3D consistency, and, consequently, also for the computation of Lax pairs, but the explicit expression (6.5) is not needed. Applying the Alternate Way, discussed in the previous section, to the B-2 system one obtains

F F F 1 000 f ψa = g = g = 0 010   := ψc, (6.61)       g H h yF + xf y x 0 0     −  −  h         e   deﬁning a 3 4 matrix . The inverse transformation, × H e F 1 0 0 y 1 F f x 0 x 1 ψc =   =   g := − ψa, (6.62) g 0 1 0   HRight     h h 0 0 1         e     1 determines the right inverse,yielding the gauge-like transformation L = L − . The a H1 cHRight 1 factorizations of and − (not shown here) are similar to those for the A-2 system (see G G e e (6.45) and (6.46)).

6.3.3 C-3 System

The generalized Hietarinta C-3 system [92],

zy + x x =0, zy + x x =0, and (6.63a) 1 1 − 2 2 − G( q, b)y2z1 G( p, b)y1z2 G( a, b)x12 yz12 + z − − − − − =0, (6.63b) − − − z1 z2 − with G(ω,κ) deﬁned in (6.5). Showing how (6.63) reduces to Hietarinta’s original C-3 system requires a few steps. Using the edge equations (6.63a), yields

x z x z z(y z y z ) x = 2 1 − 1 2 = x + 1 2 − 2 1 . (6.64) 12 z z z z 1 − 2 1 − 2 86 The latter allows one to replace (6.63b) by its alternate form:

x z G( p, a)y1z2 G( q, a)y2z1 z12 G( a, b) + − − − − − =0, (6.65) − − − y y z1 z2 − where we have used (6.7) for both p and q. Selecting G( a, b) = d , G( p, a) = p3, − − − 2 − − − and G( q, a)= q3 yields Hietarinta’s C-3 system in [64, p. 96] with d already eliminated. − − − 1 The Lax matrix L for this special case (with d = 0) is given in [64, p. 97] with references 1 6 to earlier papers where it had been derived. In [17] and [20], p3 and q3 are identiﬁed with p and q, respectively.

System (6.63) also has gauge-equivalent Lax matrices La and Lb resulting from the two ways of handling the edge constraint,

zy + x x =0, (6.66) 3 3 − which can either be solved for x3 or y3. However, the components of x13, namely,

x z x z x x x = 3 1 − 1 3 , y = 1 − 3 , and (6.67a) 13 z z 13 z z 1 − 3 1 − 3 G( a, b)(x z x z )+ z(G( k, b)y z G( p, b)y z ) z = − − 3 1 − 1 3 − − 3 1 − − − 1 3 , (6.67b) 13 y(z z ) 1 − 3 will not result in a Lax matrix Lc if one simply uses the substitutions (6.30). Subtracting zy + x x = 0 and zy + x x = 0 yields x x = z(y y ) which 1 1 − 3 3 − 1 − 3 3 − 1 allows one to replace y13 in (6.67a) by z(y y ) y = 3 − 1 . (6.68) 13 z z 1 − 3 We will now proceed with the latter expression for y13 for consistency of the derivation.

The derivation of La and Lb is similar to the A-2 system. Here again, we disregard the explicit expressions for G( p, b), G( q, b), and G( a, b). Solving (6.66) for x and − − − − − − 3

87 T using ψa = F gh , yields h i z 0 1 − 1 1   L = zy z 0 , (6.69) a z  1 −       G xz1   ( a, b) ℓ32 ℓ33 − − − y    where ℓ = G( a, b) G( k, b) zz1 , and ℓ = G( a, b)x + G( p, b)zy 1 . 32 − − − − − y 33 − − 1 − − 1 y T Similarly, solving (6.66) for y3 and using ψb = Ffh , yields h i z 0 1 − 1 1   L = 0 z x , (6.70) b z  − 1 1       G xz1 ˜ ˜   ( k, b) ℓ32 ℓ33 − − − y    where ℓ˜ = G( a, b) G( k, b) z1 , and ℓ˜ = G( a, b)x + G( p, b)zy 1 . 32 − − − − − − y 33 − − 1 − − 1 y The matrices L and L are gauge equivalent, as in (6.33), for shown in Table 6.3. a b G 1 Initially ignoring (6.66) and using the substitutions (6.30) with s = t = z , one obtains the 4 4 Lax matrix given in [92], × z 0 01 − 1   0 z 0 x  1 1  1  −  Lc =   , (6.71) z   zy1 0 z 0   −       G z1 G zz1   0 ( a, b) ( k, b) ℓ44  − − − y − − − y    G G 1 where ℓ44 = ( a, b)x1 + ( p, b)zy1 y . By (6.63a), x1 can be replaced by x zy1 or − − − − T − zy can be replaced by x x . As before, ψ = Ffgh . 1 − 1 c In contrast to the way gauge equivalences forh the A-2i and B-2 systems were dealt with in Sections 6.3.1 and 6.3.2, respectively, we are forced to use the Alternate Way discussed in

88 Section 6.3.1. Indeed,

F F F 1 0 00 xF f x 1 f ψa = g = − = 0 0   := ψc, (6.72)    z   z − z  g H h h 0 0 01         h         e   deﬁnes a 3 4 matrix which has a 3 parameter family of right inverses, × H − e 1 0 0

1 x z 0 − =  −  , (6.73) HRight,all α β γ   0 0 1 e     where α,β, and γ are free parameters.

1 Again, the inverse transformation, ψ = − ψ , requires that α = γ = 0 and β = 1. c HRight a The gauge-like transformation is of the form (6.51) with as deﬁned in (6.72) and e H 1 0 0 e

1 x z 0 − =  −  . (6.74) HRight 0 1 0   0 0 1 e     As with the A-2 system, veriﬁcation of 3D consistency and the derivation of Lax pairs for

(6.63) does neither require the explicit form nor any conditions on G(ω,κ), unless ℓ33 in (6.69) is replaced by (G( a, b)x + G( p, a)zy ) 1 and/or ℓ˜ in (6.70) by (G( a, b)x + − − − − 1 y 33 − − G( p, a)zy ) 1 . − − 1 y

Note that the matrix Lc does not have the tell-tale zero column indicating an unnecessary variable (either x3 or y3 per (6.66)). For the C-3 case, La or Lb are simply not submatrices

1 of L . Note that comes from − after removing the third row and also from after c G HRight J 1 removing the (redundant) second column. − comes from after removing the (redundant) e G H e 1 third column and also from − after removing the second row. JRight e The results for the C-3 systeme are presented in Table 6.3.

6.3.4 C-4 System

As presented in [92], the C-4 system is given by

89 zy + x x =0, zy + x x =0, and (6.75a) 1 1 − 2 2 − G(p)y z G(q)y z yz z 1 2 − 2 1 xx + 1 G( a, b)2 =0, (6.75b) 12 − z z − 12 4 − − 1 − 2 where G(τ) := 1 G( τ, a)+ G( τ, b) , and with G(ω,κ) deﬁned in (6.5). − 2 − − − − Selecting G( a, b)2 = 4d , G(p)= p3, and G(q)= q3 yields Hietarinta’s C-4 system − − − 0 − − in [64, p. 96]. In [17] and [20], p3 and q3 are identiﬁed with p and q, respectively, and the sign of x is reversed. The derivation of Lax matrices for (6.75) parallels that for the C-3 system, disregarding the expressions for G(p), G(q), and G( a, b). − − The results are presented in Table 6.4. Again, we were able to derive gauge-equivalent Lax matrices by solving the edge constraint,

x z y x =0 (6.76) 3 − 3 − for either x or y , resulting in the 3 3 matrices L and L , respectively. Leaving x and 3 3 × a b 3 y unconstrained yields the 4 4 matrices L already given in [92]. 3 × c As with the A-2 and C-3 system, veriﬁcation of 3D consistency and the derivation of Lax pairs for (6.75) does not require the form or any conditions on G(ω,κ). As with the C-3

1 system, comes from − after removing the third row and also from after removing G HRight J 1 the (redundant) second column. − comes from after removing the (redundant) third e G H e 1 column and also from − after removing the second row. JRight e e

90 Table 6.1: Generalized Hietarinta A-2 System

zx1 y1 x =0,zx2 y2 x =0, and − − − − Note: Lax pair is valid for any G(ω,κ). [92, eq. 44] G( p, a)x1 G( q, a)x2 y x z12 + b0x + − − − − − =0. − ∗ z2 z1 −

Substitutions ψ Matrix L ofLaxpair Gaugetransformations Then, for zz1 0 z The compatibility yields − x x x z x z 1   1 0 0 1 3 1 3 3 1 La = xz1 z1 zx1 , x13 = − , y13 = − , and z − z1 z3 z1 z3    x 1  − −  G −k,−a y b x z  = 0 (y + b x)(z z )+ G( k, a)x G( p, a)x F  ℓ ( ) ( + 0 )  G z z 0 1 3 3 1  31 x x    z13 = − − − − − − . ψ = g −   x(z1 z3) a where,    0 0 1 − h   y3 + x   −1 The edge constraint, x = , yields 1 we have Lb = 1La and ψb = ψa. 3   ℓ31 = G( p, a)zx1 G( k, a)x G G G z x − − − − − xF + g g h x3 = , y3 = , z3 = . (y + b0x)zz1 . zF F F − Then, for The edge constraint, y = zx x, yields z1 0 1 3 3 − − 1 0 0 f zf xF h   x3 = , y3 = − , z3 = . L = x1 1 0 , 1 000 F F F b −  x 1 0 F   z z −  G(−k,−a) y+b0x  1   ψ = f  ℓ31  =   , Left = x z 0 0 , b    − x x  H   H − h where,   0 1 0            0 001 1  0 0 1   ℓ31 = G( p, a)x1 (y + b0x)z1 .   x − − −   −1 we have Lc = 1La and ψc = ψa. H HLeft H

Then, for f g h Substitutions: x3 = , y3 = , z3 = . z1 0 01 F F F − 1 00   1 0 0 0 x1 1 00 1 −  0 10 F Lc =  , −1   z   =   , Left = 0 1 0 0 , f  0 z1 0 x1  J   J ψc =    −   x z 0   g   −     G(−k,−a) y+b0x    0 0 0 1 h  ℓ41 0     − x x   0 01     where,       −1 we have Lc = 1Lb Left and ψc = ψb. 1 J J J ℓ41 = G( p, a)x1 z1(y + b0x) . x − − −

91 Table 6.2: Generalized Hietarinta B-2 System

xx1 z1 y =0,xx2 z2 y =0, and − − − − G Note: Lax pair is valid only if G( p, k) G( q, k)= G( p, q). [92, eq. 50] ( p, q) − − − − − − − y12 + α1 + z + α2(x12 x) xx12 + − − =0. − − x2 x1 −

Substitutions ψ Matrix L ofLaxpair Gaugetransformations

The compatibility yields y xx1 0 1 − Then, for y1 y3 x3y1 x1y3 1 x13 = − , z13 = − , and La =  ℓ21 x(x α2) α2x α1 z, 1 0 0 x1 x3 x1 x3 x − − − − −   (α x α z)(x x )  y 1  2 1 1 3  yy1 xx1 y1  = 0 y13 = − − −  − −  G x x x1 x3 F     − ψ = g where,   (x α2)(y1 y3)+ G( p, k) a  0 1 0  + − − − − . h   ℓ21 =(α2x α1 z)(y xx1)   −1 x1 x3 we have L = 1La and ψ = ψa. −   − − − b G G b G y + z3 G( p, k)x xy1(x α2). The edge constraint, x3 = , yields − − − − − x yF + h g h x3 = , y3 = , z3 = . xF F F Then, for The edge constraint, z3 = xx3 y, yields, x1 1 0 − − f g xf yF 1 0 0 x3 = , y3 = , z3 = − . Lb =  y1 0 1 , 1 000 F F F −  y 1  F   x 0 x  ℓ α x α z x α  −1   ψ = f  31 2 1 2 =   , Left = 0 010 , b    − − −  H   H h where,  0 1 0       y x 0 0     −   0 0 1    ℓ31 = x1(α2x α1 z)   − − −   −1 we have Lc = 1La and ψc = ψa. H HLeft H + y1(x α2)+ G( p, k) . − − − Then, for

f g h x1 1 00 Substitutions: x3 = , y3 = , z3 = . − 1 0 0 F F F 1 0 0 0   y1 0 10  0 1 0 − − Lc =  , = , 1 = 0 1 0 0 , F     Left  ℓ31 α2x α1 z x α2 0 J  0 0 1 J f  − − −      ψc =       0 0 1 0 g        0 y1 x1 0  y x 0   h where, −  −      −1   we have Lc = 1L and ψc = ψ . J bJLeft J b ℓ31 = x1(α2x α1 z) − − − + y1(x α2)+ G( p, k) . − − −

92 Table 6.3: Generalized Hietarinta C-3 System

zy1 + x1 x =0, zy2 + x2 x =0, and − − Note: Lax pair is valid for any G(ω,κ). [92, eq. 52] G( q, b)y2z1 G( p, b)y1z2 G( a, b)x12 yz12 + z − − − − − =0. − − − z1 z2 −

Substitutions ψ Matrix L ofLaxpair Gaugetransformations

The compatibility yields zz1 0 1 Then, for x3z1 x1z3 x1 x3 x13 = − , y13 = − , and 1   1 0 0 La = zy1 z 0 , z1 z3 z1 z3 z − − G −   (x3z1 x1z3) ( a, b)  G xz1  = x z 0 z13 = − − − F  ( a, b) y ℓ32 ℓ33 G − y(z1 z3) − − −    − ψa = g where,   0 0 1 z(G( k, b)y3z1 G( p, b)y1z3)     + − − − − − . h   − we have L = L 1 and ψ = ψ . y(z1 z3)   zz1 b 1 a b a − ℓ32 = G( a, b) G( k, b) , G G G − − − − − The edge constraint, x3 = x zy3, yields y − 1 xF zg g h ℓ33 = G( a, b)(x zy1)+ G( p, b)zy1 , x3 = − , y3 = , z3 = . y − − − − − F F F z1 0 1 Then, for x x3 − The edge constraint, y3 = − , yields z 1   Lb = 0 z1 x1 , 1 0 0 f xF f h z − x = , y = − , z = .   1 0 00 3 3 3  G xz1    F zF F F  ( k, b) y ℓ32 ℓ33 x z 0 − − −   x 1  −1 − ψ = g   = z z 0 0 , Right =   , b   where, H − H   h   0 0 0       zz1 G G  0 0 01   ℓ32 = ( a, b) ( k, b) , and     y − − − − −   0 0 1 −1   1 we have La = 1Lc Right and ψa = ψc. ℓ33 = G( a, b)x1 + G( p, b)zy1 H H H y − − − − z1 0 01 By imposing the edge constraints, − Then, for x1 = zy1 + x and x3 = zy3 + x   0 z1 0 x1 x x y y − 1 0 0 3 1 3 1 Lc =  , onto y13 = − we have, y13 = z − ,   10 0 0 z1 z3 z1 z3 zy1 0 z 0  −  0 1 0 − − F   − f g h   = x 0 z 0 , 1 = , with substitutions, x3 = , y3 = , z3 = . f  0 z1 G( a, b) ℓ ℓ  Right   F F F ψc =    y 43 44 J − J  x 1  g  − − −     z z 0      −  h where, 00 0 1        0 0 1   zz1   ℓ43 = G( k, b) , and −1   we have L = 1Lc and ψ = ψc. − − − y b J JRight b J 1 ℓ44 = G( a, b)x1 + y1zG( p, b) y − − − −

93 Table 6.4: Generalized Hietarinta C-4 System

1 zy1 + x1 x =0, zy2 + x2 x =0, and G G − − G(τ) := ( τ, a)+ ( τ, b) . G G − 2 − − − − [92, eq. 57,61] (p)y1z2 (q)y2z1 1 G 2 yz12 z − xx12 + ( a, b) =0, Note: Lax pair is valid for any G(ω,κ). − z1 z2 − 4 − − −

Substitutions ψ Matrix L ofLaxpair Gaugetransformations The compatibility yields z1 0 1 Then, for x3z1 x1z3 x1 x3 − x = − , y = − , and 13 13 1 x x1 z 0 z1 z3 z1 z3 La =  , 1 0 0 − − − − 1 2 z x(x3z1 x1z3) G( a, b) (z1 z3) zz1 G = x z 0 − − 4 − − −  ℓ31 y x + (k) ℓ33   z13 = F   G − y(z1 z3) where,   0 0 1 − ψa = g   z(G(k)y z G(p)y z ) h   −1 3 1 1 3 " # we have Lb = 1La and ψb = ψa. − . z1 2 1 G 2 G G G − y(z1 z3) ℓ31 = x ( a, b) , and − − y − 4 − − The edge constraint, x3 = x zy3, yields − 1 1 2 ℓ33 = xx1 G( a, b) G(p)y1z . xF zg g h y − 4 − − − x3 = − , y3 = , z3 = . F F F x x3 The edge constraint, y3 = − , yields z z1 0 1 − Then, for f xF f h 1 x3 = , y3 = − , z3 = . 0 z1 x1 Lb =  − , 1 0 0 F F F z 1 0 00 z1  ℓ31 x + G(k) ℓ33 x z 0 y x −1   F  −  =  1 0 0 , = − , where,   H z − z HRight ψb = f 0 1 0 "h#  0 0 01   z1 1 2   0 0 1 ℓ31 = G( a, b) + G(k)x ), and     y 4 − − −1   we have L = 1Lc and ψ = ψc. b H HRight b H 1 1 2 ℓ33 = xx1 G( a, b) G(p)y1z . y − 4 − − −

By imposing the edge constraints, z1 0 0 1 − Then, for x = x zy and x = x zy 0 z 0 x 1 1 3 3  1 1  − − L = − , 1 0 0 x1 x3 y3 y1 c onto y = − we have, y = z − , zy1 0 z 0 10 0 0 13 13  −  0 1 0 z1 z3 z1 z3 F  xz1 zz1  −1   − −  ℓ41 G(k) ℓ44 = x 0 z 0 , = , f g h f  − y y    Right x 1 ψc =   J − J 0 with substitutions, x3 = , y3 = , z3 = .  g  where,  z − z  F F F h 00 0 1      0 0 1   G 2 z1   ℓ41 = ( a, b) , and −1   − − 4y we have L = 1Lc and ψ = ψc. b J JRight b J 1 1 2 ℓ44 = xx1 G( a, b) G(p)y1z . y − 4 − − −

94 CHAPTER 7 SOFTWARE IMPLEMENTATION

The CAC property has been used to identify integrable P∆Es [16, 20]. As discussed in Chapter 5, the information gained from verifying CAC is also crucial for the computation of the corresponding Lax pair. In some sense the lattice equation is its own Lax pair (cf. the discussion in [38]). For scalar P∆Es, CAC is a simple concept that can be verified by hand or (interac- tively) with a computer algebra system (CAS) such as Mathematica or Maple. Hereman [19] designed software to compute Lax pairs of scalar P∆Es defined on a quadrilateral with the CAC property. For systems of P∆Es with edge equations the verification of the CAC property can be tricky and the order in which substitutions are carried out is important. Designing a symbolic manipulation package that fully automates the steps has been quite a challenge [17].

7.1 Algorithm

Naively, one could ﬁrst generate the comprehensive system that represents the P∆Es on each face of the cube and then ask a CAS to solve it. To be consistent around the cube, that system should have a unique solution for x123. Wolf [94] discusses the computational challenges of verifying the CAC property for scalar P∆Es in 3 dimensions [95] due to the astronomical size of the overdetermined system that has to be solved. Even for P∆Es in two dimensions, in particular, those involving edge equations, automatically solving such a system often exceeds the capabilities of current symbolic software packages. It is therefore necessary to verify CAC in a more systematic way like one would do with pen on paper. Computer code [17] for automated veriﬁcation of the CAC property carries out the following steps:

95 1. Solve the initial P∆E for x12. Solve the equations on the bottom and left faces for x13

and x23, respectively. Generate the equations for the back, right and top equations and

solve each for x123. This produces three independent expressions for the components

of x123.

2. Evaluate and simplify the solutions x123 using x12, x13, and x23. Use the constraints

between the components of x, x1, x2, and x3 arising from the edge equations to check consistency at every level of the computation.

3. Finally, verify if the three expressions for the components of x123 are indeed equal. If so, the system of P∆Es is consistent around the cube and one can proceed with the computation of a pair of Lax matrices.

7.1.1 Computation of a Lax pair

Assuming the given P∆E is CAC, the following steps are then taken to calculate a Lax pair:

f g 1. Introduce fractional expressions (e.g., F , G , etc.) for the various components of x3 in

order to linearize the numerators and denominators of the expressions for x13 in terms of f,F,g,G, etc., which are called projective coordinates14.

2. Further simplify the components of x3 using the edge equations (if present in the given P∆E).

3. Substitute the simpliﬁed expressions for x3 into x13 and again examine if the numerators and denominators are linear in f,F,g,G, etc.

4. If x13 is not yet “linearized”, reduce the degree of freedom (e.g., by setting G = F, etc.) and repeat this procedure until the numerators and denominators of the components

of x13 are linear in f,F,g, etc.

14These fractional transformations are used in the same spirit as those for linearization of (continuous) Riccatti equations.

96 5. Use the fractional linear expressions of x13 to generate the “core” Lax matrix, Lcore.

6. Use the determinant method (see Section 5.3) to compute a possible scaling factor t.

7. The Lax matrix is then L = tLcore. The matrix M = sMcore follows from L by replacing

p by q and x1 by x2, leaving the components of x and k the waythey are.

7.1.2 Veriﬁcation of the Lax pair

Finally, verify the Lax pair by substitution into the Lax equation (2.18). Unfortunately, the determinant method gives s and t in irrational form, introducing, e.g., square or cubic roots into the symbolic computations. In general, symbolic software has limited capabilities for simplifying expressions involving radicals due to possible branching and the face that symbolic software does not replace e.g, √a2 by a , unless it is instructed to do so. Similarly, | | as exempliﬁed below, √3 a does not automatically simplify to a because computer algebra systems can not assume that a is real. The impact of the presence of radical expressions can be reduced by careful simpliﬁcation. Notice that (5.12) can be written as

(st2) (Lcore)2Mcore (Mcore)1Lcore ˙=0. (7.1a) (ts1) − Bringing all common factors from the matrix products up front gives

st CF 2 L2M L˜ M˜ M˜ L˜ ˙=0 (7.1b) ts CF 2 − 1 1 M1L ˜ ˜ where CFX stands for a common factor of all the entries of a matrix X. Hence, CFL2M L2M = ˜ ˜ (Lcore)2Mcore and CFM1L M1L =(Mcore)1Lcore. The computed Lax pair is correct if st CF 2 L2M ˙= 1 (7.2a) ts CF ± 1 M1L and, thus

L˜ M˜ M˜ L˜ ˙=0. (7.2b) ± 2 − 1

97 To illustrate the veriﬁcation procedure, consider Example 2 with t and s in (5.47). Here,

x2(y y )3(z z ) 3 x 3 2 1 2 2 − − 2 st (k q)y2 (z z2) (k p)yy2(py2(z1 z)+qy1(z z2)) (z1 z2) 2 = − − − − − − , (7.3a) q q x2(y y )3(z z ) ts1 3 x 3 2 1 1 2 − − 2 (k p)y (z z1) (k q)yy1(py2(z1 z)+qy1(z z2)) (z1 z2) − 1 − − − − − q y q y CF = 2 and CF = 1 . (7.3b) L2M x(y y ) M1L x(y y ) 1 − 2 1 − 2 The matrix L˜2M˜ (which equals M˜1L˜) is

pqy(z z ) ky(qy py ) ky(py z qy z ) − 1 − 2 1 − 2 2 1 − 1 2   pz (z z )+ qz (z z) k(y z y z ) kz(y z y z )  2 1 1 2 1 2 2 1 2 1 1 2   − − − −   + py2(z1 z)+ qy1(z z2)  .  − −         p(z z1)+ q(z2 z) k(y1 y2) kz(y2 y1)+ py2(z1 z)  − − − − −   + qy1(z z2)   −   (7.4) Note that CF y L2M = 2 . (7.5) CFM1L y1 After multiplying (7.5) with (7.3a), the resulting expression can be simpliﬁed into 1. Thus, both (7.2a) and (7.2b) are satisﬁed for the plus sign.

7.2 Software Package

The algorithm presented in Chapter 5 and summarized above, was implemented in a Mathematica package, LaxPairPartialDifferenceEquations.m, as part of the research for this dissertation. This development extended the work done by Hereman [19] for the computation of a Lax pair of scalar lattice equations, to include computation of a Lax pair for systems of equations. In short, the software will determine if a speciﬁed P∆E, or system of P∆Es, is CAC and if so, compute a corresponding Lax pair. Additionally, for a given Lax pair and P∆E, the software will determine if the Lax pair satisﬁes the governing Lax equation (2.18).

98 The software targets users who wish to test or confirm the integrability of a given P∆E. The user may specify a P∆E in an external data file and then have the software compute a Lax pair. If the user has a candidate Lax pair for the given P∆E, they may use the software to verify if that Lax pair satisfies the Lax equation. Before running the code, the user should place the program file and all data files provided with the software into one directory. For convenience, a Mathematica notebook has been provided containing the necessary commands to run LaxPairPartialDifferenceEquations.m. As indicated in the sample data file in Section 7.2.2, the following commands are necessary for executing the software package:

• SetDirectory[ NotebookDirectory[] ]

Establishes the current working directory as the directory where the notebook resides.

• Get[ "LaxPairPartialDifferenceEquations.m" ]

Loads the main code with the functionality based on the algorithm discussed in Sec- tion 5.

• LaxPairSolverUI[ " User speciﬁed path ", userInterfaceDebugFlag->False];

Launches the interface allowing the user to specify the functionality to be executed. The user may specify any path as the User specified path, but the directory specified must contain the sampleLattice directory in order to access the data files provided with the code.

7.2.1 LPSolve

Once the software package is initialized, the user will be presented with a dialog window shown in Figure 7.1. The secondary screen is accessible via the Advanced Directives tab near the top of the dialog. The Advanced Directives tab allows the user to either suppress (by default) or allow additional debugging information presented during the course of execution. Typically, this information is suppressed as the additional information slows execution. At this point, the user has several options:

99 Figure 7.1: Initial Dialog of LPSolve

• The user may elect to process an existing sample lattice. For convenience, the available sample P∆Es are presented as Scalar Equations or System of Equations and listed in the respective drop-down menus; or

• The user may elect to process a user provided P∆E. This option requires the user to prepare a data file with the P∆E to be analyzed (for format information, see Section 7.2.2). Once a P∆E has been specified, the user may generate statistics which outputs the contents of the lattice file (for verification purposes) as well as

• equations contained,

• variables speciﬁed,

• additional parameters, if any,

100 • double-subscripted variables (essential for veriﬁcation of the CAC property),

• type of equations speciﬁed (as based on the quadrilateral, i.e., double edge equations, and single edge equations). The user may also select the type of processing desired, including

• Check Consistency Around the Cube,

• Compute a Lax Pair,

• Verify the User Provided Lax Pair, and then begin computation by selecting Process Systems. The resulting information is then printed to the Lax Pair Output window and may be reviewed and/or saved (see Section 7.2.3).

7.2.2 Sample Lattice Files

In addition to the core software, the package includes sample lattice ﬁles for all the scalar lattice equations and systems thereof that have been examined over the course of this research. In each data ﬁle the user may specify

• nameINPUT : the name of the P∆E speciﬁed; (optional)

• ddeEquationListINPUT : the equation or list of equations that constitute the P∆E speciﬁed; (required)

• laxPairMatrixL : user speciﬁed Lax pair L matrix; (optional)

• laxPairMatrixM : user speciﬁed Lax pair M matrix; (optional)

• explicitScalars : indicates that tFunc and sFunc are speciﬁed; (optional with acceptable values of True or False)

• tFunc : user speciﬁed scalar function corresponding to the L matrix; (optional)

• sFunc : user speciﬁed scalar function corresponding to the M matrix; (optional). The L and M matrices and, optionally, the sFunc and tFunc, are speciﬁed if the user plans to verify them as a valid Lax pair.

• userShifts : indicates that governing parameter constraints are speciﬁed; (optional with acceptable values of True or False)

101 • paramEquivalences : Any governing parameter relationship(s). The provided governing equations will be applied to the specified P∆E during the CAC verification, Lax pair derivation and/or user Lax pair verification process; (required, dependent upon the P∆E, for example, see Table 6.3). An example of a data file for a system of P∆Es is shown in Figure 7.2.

ddEQ = {z[0,0,0]*y[1,0,0] + x[1,0,0] - x[0,0,0], z[0,0,0]*y[0,1,0] + x[0,1,0] - x[0,0,0], G[-a,-b]*x[1,1,0] - y[0,0,0]*z[1,1,0] + z[0,0,0]*((G[-q,-b]*y[0,1,0]*z[1,0,0] - G[-p,-b]*y[1,0,0]*z[0,1,0])/(z[1,0,0] - z[0,1,0]))};

nameINPUT = "Generalized C-3 (per Zhang, Zhou, Nijhoff)"; ddeEquationListINPUT = ddEQ;

laxPairMatrixL = {{-z[1,0,0], 0, 1}, {z[0,0,0]*y[1,0,0], -z[0,0,0], 0}, {-G[-a,-b]*(x[0,0,0]*z[1,0,0])/y[0,0,0], (G[-a,-b] - G[-k,-b])*(z[0,0,0]*z[1,0,0])/y[0,0,0], (G[-a,-b]*x[1,0,0] + G[-p,-b]*z[0,0,0]*y[1,0,0])/y[0,0,0]}};

laxPairMatrixM = {{-z[0,1,0], 0, 1}, {z[0,0,0]*y[0,1,0], -z[0,0,0], 0}, {-G[-a,-b]*(x[0,0,0]*z[0,1,0])/y[0,0,0], (G[-a,-b] - G[-k,-b])*(z[0,0,0]*z[0,1,0])/y[0,0,0], (G[-a,-b]*x[0,1,0] + G[-p,-b]*z[0,0,0]*y[0,1,0])/y[0,0,0]}};

explicitScalars = True;

tFunc = 1/z[0,0,0]; sFunc = 1/z[0,0,0];

userShifts = True; paramEquivalences = {G[-p, -q] -> G[-k,-q] - G[-k,-p], G[-p,-k]-> -G[-k,-p]};

Figure 7.2: Sample data ﬁle for the generalized C-3 equation

7.2.3 Output Data Files

The intermediate calculations and ﬁnal results are presented in a separate output window along with any issues/errors encountered during the computation (see Figure 7.3 and

102 Figure 7.4)15. This output may be saved for each computation separately or appended to with results of additional computations. The software has been used to compute Lax pairs of P∆Es presented in this dissertation (see the various tables provided throughout). The data ﬁles for these equations are provided as part of the software.

Figure 7.3: Sample output of LPSolve for the Q1 equation

Note that, as part of each output file, the specified P∆E is given as well as the processing (i.e., CAC verification, Lax pair derivation, etc.) specified by the user. Figure 7.4 shows output for the Boussinesq system which has two edge equations and one equation defined on the full square. Note that the specified system of P∆Es is augmented with additional edge equations to ‘complete’ the system prior to processing.

15Note: Only a portion of the actual output ﬁle is shown. Since the user has the option for three individual processes, these ﬁles can be quite long.

103 Figure 7.4: Sample output of LPSolve for the Boussinesq system of equations

104 CHAPTER 8 CONCLUSIONS AND FUTURE DIRECTION

The purpose of the research conducted for this dissertation was twofold. On the theoretical side, we did a literature review to find some of the origins of the P∆Es discussed in this dissertation. This lead us to investigate Bäcklund transformations, Hirota’s discrete bilinear operators, and direct linearization techniques for both continuous and discrete equations, and continuum limits. On the software side, the main purpose of this dissertation was to extend the algorithm and code used to compute Lax pairs for scalar P∆Es (see [19]) to systems of P∆Es. The implementation of that algorithm is a major component of the novel research presented in this thesis. The algorithm has been implemented in Mathematica, a commonly used computer algebra system. Our symbolic package, LaxPairPartialDifferenceEquations.m which accompanies the thesis, allows the user to automatically (i) determine whether or not P∆Es have the CAC property; (ii) compute Lax pairs for nonlinear P∆Es that are CAC; and (iii) verify if Lax pairs satisfy the Lax equation. In this thesis, Lax pairs are presented for the scalar integrable P∆Es discovered by Adler, Bobenko, and Suris as well as for numerous systems of integrable P∆Es, including the lattice Boussinesq, Schwarzian Boussinesq, Toda-Modified Boussinesq systems, and the two-component potential Korteweg-de Vries system. Using the software designed for this thesis, previously unknown Lax pairs were computed for systems of P∆Es discovered by Hietarinta. The study of Hietarinta’s systems also lead us to an investigation of gauge and gauge-like equivalence which was not reported in the literature.

8.1 Future Directions and Open Questions

Several questions have arisen during the historical research into the origins of the P∆Es as well as during the analysis of systems of P∆Es treated in this dissertation. In the context

105 of continuum limits, it is well known that the H1 and H3 equations correspond to PDEs in the family of KdV equations (see Chapter 4). Also, it has been shown [38] that the Q4 equation corresponds to the Krichever-Novikov PDE. An interesting question is what PDEs correspond to the remaining P∆Es of the ABS classification. Another interesting question is which Bäcklund transformations give rise to P∆Es on the quadrilateral. The tetrahedron property as an attribute of a P∆E being consistent around the cube, has received considerable discussion, primarily as it is an unusual assumption at first glance. With the advent of systems of P∆Es being CAC, we have encountered several systems which satisfy the tetrahedron property in one or perhaps two variables, but not necessarily in all variables. The impact of not having the tetrahedron property for all variables remains unclear. Also, in the context of construction of a Lax pair, the 2-component lpKdV equation (see Table 5.2) remains an anomoly in that it is the only system addressed thus far that requires two scalar functions in the creation of a Lax pair. As such, it is the only system thus far whose Lax pair has a larger dimension that 3 3. The question arises as to whether × other systems exist which have similar properties or result in larger dimensional Lax pairs. In the context of gauge-equivalence, the derivation of gauge matrices for the A-2 and B-2 systems was straightforward and could be done in two ways. Indeed, for both systems there was some freedom in using left and right inverses to construct the gauge-like transformations between Lax matrices of different sizes. However, for the C-3 and C-4 systems, the approach with the left inverses does no longer work. The reason for that will require further investigation. Also, the discovery of the gauge-like relationship between Lax pairs implies a greater freedom on the formulation of Lax pairs than first assumed. Questions concerning the existence of like systems or additional, higher-dimensional Lax pairs for known P∆Es remain to be answered. In the PDE case, application of the inverse scattering transform (IST) is easier if one selects a Lax pair of a specific form (i.e., the eigenvalues should appear in the diagonal entries), chosen from the infinite number of gauge equivalent pairs. Thus, for the KdV

106 equation one may prefer to work with (2.16) instead of (2.10). Similar issues arise for P∆Es. Among the family of gauge-equivalent Lax pairs for P∆Es, which one should be selected so that, for example, the IST or staircase method [44] could be applied? (The latter method is used to ﬁnd ﬁrst integrals for periodic reductions of integrable P∆Es). In addition, one

has to select an appropriate (separation) factor t(x, x1; p,k) (see Section 6.3). These open questions will require further study. With regard to the development of the software, the advent of the generalized Hietarinta systems [92] brought forth many issues still to be addressed. In addition to the questions related to gauge equivalence, these systems contain constrained parameters. As in the case of the generalized C-3 equation (see Table 6.3), the constraint was necessary for confirma- tion of CAC and computation of a Lax pair. An interesting question is whether defined constraints for the coefficients used in the Q4 equation would reduce the complexity and allow the software package to compute corresponding Lax pairs. Additionally, covering systems with arbitrary parameters and offering the user the ability to define constraints based on intermediate calculations, would require further development of software tools. Future research will include an investigation of such enhancements so that the software can handle parameterized P∆Es. In turn, the analysis of systems with parameters may lead to the discovery of new integrable P∆Es.

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116 APPENDIX A LINEARIZING THE LPKDV EQUATION

The lpKdV equation,

(p + q u + u )(p q + u u )= p2 q2, (A.1) − n+1,m+1 n,m − n,m+1 − n+1,m −

may be linearized by setting un,m = ερn,m i.e.,

(p + q)(ρn,m+1 ρn+1,m)ε (p q)(ρn+1,m+1 ρn,m)ε − − − − (A.2) (ρ ρ )(ρ ρ )ε2 =0, − n+1,m+1 − n,m n,m+1 − n+1,m and taking the terms linear in ε (ignoring terms in ε2), yielding

(p + q)(ρ ρ )=(p q)(ρ ρ ). (A.3) n,m+1 − n+1,m − n+1,m+1 − n,m As outlined in [64, p. 138], just as the originating equation, (A.1), is CAC, a simple calculation shows that the linearized equation, (A.3), is also CAC. As such (A.3) can be embedded consistently into a multi-dimensional lattice,

(p + q)(x x )=(p q)(x x), (A.4a) 2 − 1 − 12 − (p + k)(x x )=(p k)(x x), and (A.4b) 3 − 1 − 13 − (k + q)(x x )=(k q)(x x), (A.4c) 2 − 3 − 23 −

where k is the lattice parameter in the third direction. If we assume that the lattice parameter k associated with the mapping x k x is such that x = 0, then (A.4) can be solved, giving, 7−→ 3 3 p k q k x = − x, and x = − x, (A.5) 1 p + k 2 q + k As ρn,m = x, ρn+1,m = x1 and ρn,m+1 = x2, by iteration, we have

p k n q k m ρ = − − ρ , (A.6) n,m p + k q + k 0,0 16 where ρ0,0, an arbitary, constant initial solution, is a solution to (A.3) .

16Some authors use k instead of k, as can be seen in the Chapter 3 discussion regarding continuum limits. −

117 APPENDIX B LPKDV EQUATION TO LKDV EQUATION

In this appendix we will show how four P∆Es which occur in several chapters of this dissertation can be transformed into each other.

Theorem 2. The lattice pKdV equation,

(p q + w w )(p + q + w w )= p2 q2, (B.1) − n,m+1 − n+1,m n,m − n+1,m+1 − can be transformed into its simpler form,

(u u )(u u )= p2 q2. (B.2) n,m+1 − n+1,m n,m − n+1,m+1 − The latter equation can be transformed into the double pKdV equation,

un+1,m un,m+1 un,m 1 + un 1,m − − − − 1 1 =(p2 q2) , (B.3) − un 1,m 1 un,m − un,m un+1,m+1 − − − − which, in turn, can be transformed into the lattice KdV equation,

1 v v =(p2 q2) . (B.4) n+1,m − n,m+1 − v v n+1,m+1 − n,m The last equation arises in early work by Hirota [5]. In the four equations above p and q are arbitrary constants. To match similar equations in the literature, see for example [64], one has to occasionally swap p and q. To get a normalized form one can multiply the ﬁeld variable u (and its shifts) by p2 q2. Doing so would allow one to replace p2 q2 in n,m − − the right hand sides of the last threep equations by 1.

Proof.

To go from (B.1) to (B.2), set

wn,m = un,m + pn + qm + c, (B.5)

118 where c is an arbitrary constant. Substitution of (B.5) into (B.1) gives (B.2). To get (B.3) from (B.2) requires two steps. First, (B.2) can be written as

p2 q2 u u = − , (B.6) n,m+1 − n+1,m u u n,m − n+1,m+1 which, after replacement of n by n 1 and m by m 1, yields − − p2 q2 un 1,m un,m 1 = − . (B.7) − − − un 1,m 1 un,m − − − Second, subtract the ﬁrst equation from the second to get (B.3). To obtain (B.4), deﬁne

v := u u (B.8) n+1,m+1 n,m − n+1,m+1 and replace all instances of un,m (and its shifts) in (B.3) in terms of vn,m (and its shifts). Doing so, (B.3) becomes (B.4).

Note that (B.8) is a diﬀerence across the diagonal which is commonly used [64, p. 103] as one of the (many) alternatives to discretize the forward diﬀerence operator ∆ := D D I, n m −

where Dn and Dm are the forward shift operators in the n and m directions, respectively, and I is the identity operator. Hence, f =(D D I)f = f f . △ n,m n m − n,m n+1,m+1 − n,m Also note that (B.8) has two copies of the pKdV lattice. Therefore, it is called the double lattice pKdV equation [96]. Obviously, any solution of (B.5) is a solution of (B.3) but the converse statement is not necessarily true. In fact, every solution u of (B.3) can be written [96] as u = U + V, where U is a solution of (B.7) and V is a solution of the linear equation

Vn+1,m+1 = Vn.m. Obviously, (B.3) is more general than the pKdV equation (B.2). As a matter of fact, (B.3) is the Euler-Lagrange equation corresponding to an action integral of a suitable Lagrangian function (L). Consider the discrete action integral [97] for (B.1),

([u ]) = L(n, [u ]) J n,m n,m n,m Z X∈ := u (u u )+(p2 q2)ln p + q + u u , (B.9) n,m+1 n+1,m+1 − n,m − | n,m − n+1,m+1| n,m Z X∈

119 ‘ where [un,m] denotes the ensemble of un,m and its forward and backwards shifts. The discrete Euler-Lagrange equation is then readily computed using the formulas given in [98]. Indeed, to extremize ([u ]) for all variations [u ]+ ǫ[h ] of the variable u J n,m n,m n,m n,m (and its shifts), compute the discrete variational derivative

δ ([u ]) d J n,m = ([u ]+ ǫ[h ]) (B.10) δu dǫJ n,m n,m n,m ǫ=0 and equate the result to zero. This yields

∂ 1 2 1 2 1 1 2 1 I + Dn− + Dn− + + Dm− + Dm− + + Dn− Dm− + Dn− Dm− + L ∂un,m ··· ··· ··· !

= un+1,m un,m+1 un,m 1 + un 1,m − − − − 1 1 +(p2 q2) =0, (B.11) − p + q + un,m un+1,m+1 − p + q + un 1,m 1 un,m − − − − ! 1 1 where Dn− and Dm− are the backward shift operators in the n and m directions, respectively. Setting

un,m =u ˜n,m + pn + qm + c, (B.12) allows one to replace (B.11) by

uñ+1,m uñ,m+1 uñ,m 1 +˜un 1,m − − − − 1 1 =(p2 q2) . (B.13) − uñ 1,m 1 uñ,m − uñ,m uñ+1,m+1 − − − − After dropping the tilde, (B.13) matches (B.3). Finally, we show how (B.3) also can be obtained from the so-called three-leg form [16] of (B.2). Adler et al. showed [16] that each of the equations of their ABS classification (see section 3.5.2) can be written in three-leg form

ψ(u ,u ; α) ψ(u ,u ; β)= φ(u ,u ; α, β), (B.14) n,m n+1,m − n,m n,m+1 n,m n+1,m+1 where the parameters α and β and the functions ψ and φ depend on the speciﬁc P∆E one is dealing with. For example, for (B.2)

120 ψ(un,m,un+1,m; α) = un,m + un+1,m, (B.15) α β φ(u ,u ; α, β) = − , (B.16) n,m n+1,m+1 u u n+1,m+1 − n,m with α = p2 and β = q2.

v2 α2 y2

β2 β2

u1 α1

x α2 u2

β1 β1

y1 α1 v1

Figure B.1: The extended lattice with ABS labeling

Then, based on the three-leg form and referencing an extended lattice shown in Figure Figure B.1, Adler et al. derived an action for each of the equations in the ABS classiﬁcation, namely,

= L(X,U; α) L(X,V ; β) Λ(X,Y ; α, β), (B.17) F − − X,U E1 X,V E2 X,Y E3 X∈ X∈ X∈ where E1 is the set of horizontal edges through x, E2 is the set of vertical edges through x, and E3 is the set of diagonal edges through x. Furthermore, they showed that there exist symmetric functions L(X,U; α) = L(U, X; α) and Λ(X,Y ; α, β) = Λ(Y,X; α, β) which are related to ψ and φ as follows: ∂ ψ(x,u; α)= ψ(f(X),f(U); α)= L(X,U; α), and ∂X (B.18) ∂ φ(x,y; α, β)= φ(f(X),f(Y ); α, β)= Λ(X,U; α, β). ∂X

121 They also showed that the corresponding Euler-Lagrange equation is then given as

ψ(x,u ; α )+ ψ(x,u ; α ) ψ(x,v ; β )+ ψ(x,v ; β ) 1 1 2 2 − 1 1 2 2 (B.19) φ(x,y ; α , β )+ φ(x,y ; α , β ) =0. − 1 1 1 2 2 2 Inserting (B.15) and (B.15), Equation (B.19) becomes

α β α β x + u + x + u x + v + x + v 1 − 1 + 2 − 2 =0. (B.20) 1 2 − 1 2 − y x y x 1 − 2 − Inserting x = un,m, u1 = un 1,m, u2 = un+1,m, v1 = un,m 1, v2 = un,m+1, y1 = un 1,m 1, − − − − 2 2 y2 = un+1,m+1, α1 = α2 = p , and β1 = β2 = q reduces (B.20) to (B.3) after simpliﬁcation.

122 APPENDIX C DIRECT LINEARIZATION OF KORTEWEG-DE VRIES EQUATION

Fokas and Ablowitz [65, 66] have shown that if ϕ(x,t; k) solves the singular linear integral equation,

ϕ(x,t; ℓ) ϕ(x,t; k)+ iρ(x,t; k) dλ(ℓ)= ρ(x,t; k) (C.1) ℓ + k ZC with plane-wave factor

3 ρ(x,t; k)= ei(kx+k t) (C.2) and where dλ(k) and C are an appropriate measure and contour, respectively, then

w(x,t)= ϕ(x,t; k)dλ(k) (C.3) ZC satisﬁes the potential KdV equation,

w 3w2 + w =0, (C.4) t − x 3x and therefore

u = w = ∂ ϕ(x,t; k)dλ(k) (C.5) − x − x ZC satisﬁes the KdV equation,

ut +6uux + u3x =0. (C.6)

Proof.

The contour and measure are both arbitrary but such that

(i) diﬀerentiation with respect to x and t and integration along the contour can be inter- changed, and

(ii) the homogeneous integral equation, i.e., (3.67) with zero on the right hand side, has only the zero solution (ϕ(x,t; k) 0). ≡

123 To simplify the notation we will use the following short scripts:

dλ(ℓ) ϕ(x,t; k) ϕ, ρ(x,t; k) ρ, , and dµ. (C.7) 7→ 7→ 7→ ℓ + k 7→ ZC Z Applying the diﬀerential operator

3 L = ∂t + ∂x +3u∂x (C.8)

to (3.67) yields

Lϕ + iL ρ ϕdµ = Lρ (C.9) Z or

3 3 Lϕ + i ∂t + ∂x +3u∂x ρ ϕdµ = ∂t + ∂x +3u∂x ρ. (C.10) Z Explicitly,

Lϕ + i ρt ϕdµ + ρ ϕtdµ + ρ3x ϕdµ +3ρxx ϕxdµ +3ρx ϕxxdµ " Z Z Z Z Z

+ ρ ϕ3xdµ +3uρx ϕdµ +3uρ ϕxdµ = ρt + ρ3x +3uρx. (C.11) Z Z Z Upon substitution of the derivatives of (C.2) and rearranging the terms, one gets

Lϕ + iρ ϕtdµ + ϕ3xdµ +3u ϕx dµ Z Z Z + i ik3ρ ϕdµ ik3ρ ϕdµ 3k2ρ ϕ dµ +3ikρ ϕ dµ − − x xx " Z Z Z Z +3ikuρ ϕdµ = ik3ρ +(ik)3ρ +3ikuρ (C.12) Z # or

2 Lϕ + iρ Lϕdµ =3ikuρ +3kuρ ϕdµ +3ik ρ ϕxdµ +3kρ ϕxxdµ. (C.13) Z Z Z Z

124 To simplify we need expressions for the three integrals appearing on the right hand side of (C.13). From (3.67),

iρ ϕdµ = ρ ϕ. (C.14) − Z After diﬀerentiation of (C.14) with respect to x one gets

iρ ϕ dµ = ikρ ϕ + kρ ϕdµ. (C.15) x − x Z Z After one more diﬀerentiation one obtains

iρ ϕ dµ = k2ρ ϕ +2ikϕ ik2ρ ϕdµ. (C.16) xx − xx x − Z Z Substitution of (C.15) and (C.16) into (C.13) gives

Lϕ + iρ Lϕdµ =3ikuρ +3kuρ ϕdµ +3k2 ikρ ϕ + kρ ϕdµ − x Z Z Z 3ik k2ρ ϕ +2ikϕ ik2ρ ϕdµ . (C.17) − − xx x − Z Hence,

Lϕ + iρ Lϕdµ =3ikuρ +3k [iϕxx + kϕx]+3kuρ ϕdµ, (C.18) Z Z which, after substitution of (C.14), simpliﬁes into

Lϕ + iρ Lϕdµ =3k [iϕxx + kϕx + iuϕ] . (C.19) Z Next, we will show that

ϕ(k)= iϕxx(k)+ kϕx(k)+ iuϕ(k) (C.20) satisﬁes the homogeneous integrale equation, i.e.,

ϕ(k)+ iρ ϕ(ℓ)dµ =0. (C.21) Z To do so, deﬁne the operator e e

2 Mk = i∂x + k∂x + iuI, (C.22)

125 which depends on k and where I is the identity operator such that

ϕ(k)= Mkϕ(k) (C.23) and apply Mk to (3.67), yielding e

Mkϕ + iMk ρ ϕdµ = Mkρ, (C.24) Z or, explicitly,

2 2 Mkϕ + i i∂x + k∂x + iuI ρ ϕdµ = i∂x + k∂x + iuI ρ(k). (C.25) Z Hence,

Mkϕ + i i ρxx ϕdµ +2ρx ϕxdµ + ρ ϕxxdµ " Z Z Z

+ k ρx ϕdµ + ρ ϕxdµ + iuρ ϕdµ = iρxx + kρx + iuρ. (C.26) Z Z Z #

Substituting for ρx and ρxx yields

M ϕ + i i k2ρ ϕdµ +2ikρ ϕ dµ + ρ ϕ dµ k − x xx " Z Z Z + k ikρ ϕdµ + ρ ϕ dµ + iuρ ϕdµ = i( k2ρ)+ ik2ρ + iuρ. (C.27) x − Z Z Z # After rearranging the terms,

Mkϕ(k)+ iρ(k) Mkϕ(ℓ)dµ = iuρ(k)+2kiρ ϕx(ℓ)dµ, (C.28) Z Z where we have denoted the dependencies on k and ℓ explicitly. Finally, the operator Mk in the integrand on the left hand side of (C.28) must be be replaced by Mℓ. Based on (C.22),

M = M +(k ℓ)∂ , (C.29) k ℓ − x and therefore (C.28) becomes (after rearrangement)

126 Mkϕ(k)+ iρ(k) Mℓϕ(ℓ)dµ = iuρ(k)+ kiρ(k) ϕx(ℓ)dµ + iρ(k) ℓϕx(ℓ)dµ, Z Z Z = iuρ(k)+ iρ(k) (k + ℓ)ϕx(ℓ)dµ Z = iuρ(k)+ iρ(k) ϕx(ℓ)dλ(ℓ) Z = iuρ(k)+ iρ(k)( u) − =0, (C.30)

where we have used (C.7) (relating dµ to dλ) and (C.5) to simplify. Hence, we have proved that (C.20) holds. Since the homogeneous integral equation has only the trivial solution, it follows that

Mkϕ(k)=0, (C.31)

that is,

2 (i∂x + k∂x + iuI)ϕ(k)=0, (C.32)

Consequently, the right hand side of (C.19) is zero, and using the same argument one has

Lϕ(k)=0, (C.33)

that is,

3 (∂t + ∂x +3u∂x)ϕ(k)=0. (C.34)

Integration of (C.33) with respect to dλ(k) over the contour C yields

(Lϕ(k))dλ(k)=0 (C.35) Z or, explicitly,

3 3 (∂t + ∂x +3u∂x)ϕ(k)dλ(k)= ∂t ϕ(k)dλ(k)+ ∂x ϕ(k)dλ(k)+3u ∂xϕ(k)dλ(k)=0. (C.36) Z Z Z Z Using (C.3) and (C.5), this can be written as

w + w 3u2 =0. (C.37) t 3x −

127 Using (C.5) to replace u, (C.37) becomes (C.4). Diﬀerentiating (C.37) with respect to x and using w = u to replace all derivatives of w gives (C.6). x −

Remarks:

(i) To proof (C.21), instead of applying the operator Mk to (3.67), one could have applied

2 operator M0 = i∂x + iuI (which is independent of k) to (3.67), rearrange the terms, use k = k+ℓ ℓ = 1 ℓ , and then add (3.67) (after multiplication with iuϕ) to the k+ℓ k+ℓ − k+ℓ − k+ℓ respective sides [99].

(ii) When the nonlinear term in (C.6) is absent then (3.67) reduces to

3 ϕ(x,t; k)= ρ(x,t; k)=ei(kx+k t) (C.38) and, therefore,

3 u = w = ∂ ei(kx+k t)dλ(k) (C.39) − x − x ZC is the most general solution of the linearized KdV equation, ut + uxxx =0.

(iii) Regardless of the proof technique, the solution ϕ and u are related via (C.5) with ϕ satisfying (3.67). As consequences of the method of proof, (C.32) and (C.34) must hold and these be written [12] as

(kI + i∂x)i∂xϕ = uϕ, (C.40)

and

(∂ + ∂3)ϕ = 3uϕ . (C.41) t x − x

(iv) As shown in [65, 66], by substituting

i (kx+k3t) ϕ = ψe 2 (C.42) into (C.32), ψ satisﬁes the Schr¨odinger equation 1 ψ +( k2 + u)ψ =0. (C.43) xx 4

128 Likewise, substitution of (C.42) into (C.41) yields

ψ (k2 2u)ψ u ψ =0. (C.44) t − − x − x Equations (C.43) and (C.44) form a Lax pair [100] for the KdV equation and their compatibility yields (C.6). The latter is readily shown by diﬀerentiating (C.43) with respect to t and (C.44) twice with respect to x, equating ψxxt and ψtxx, and using (C.43) to eliminate

ψxx and ψ3x, and (C.44) to eliminate ψt. Equation (C.44) depends explicitly on the spectral parameter k. Using the x derivative of (C.43), that spectral dependence can be removed − and (C.44) can be replaced by

ψt +6uψ +3uxψx +4ψ3x =0, (C.45)

The couple (C.43)-(C.45) is a Lax pair that is also frequently reported in the literature [45, 48].

(v) Integral equation (3.67) can be made [100] a bit simpler at the cost of making (C.5) slightly more complicated. Indeed, substitution of

ϕ = ρφ (C.46) into (3.67) and (C.5) yields

φ(x,t; ℓ) 3 φ(x,t; k)+ i ei(ℓx+ℓ t)dλ(ℓ)=1 (C.47) ℓ + k ZC and

3 u = ∂ φ(x,t; k)ei(kx+k t)dλ(k). (C.48) − x ZC

129 APPENDIX D HIROTA’S BILINEAR OPERATOR

D.1 Hirota’s Bilinear Diﬀerential Operators

Let f(x) and g(x) be continuously diﬀerentiable functions to all orders in x. Hirota’s bilinear operator [72], Dx, acting on f(x) and g(x) is deﬁned by ∂ ∂ Dx(f g) := f(x)g(x′) . (D.1) · ∂x − ∂x ′ ′ x =x

The following expressions may readily be derived from this d eﬁnition,

∂ ∂ ′ Dx(f g)= f(x)g(x′) = fx(x)g(x′) f(x)gx (x′) x′=x · ∂x − ∂x ′ − | ′ x =x

=fxg fgx, (D.2) −

2 2 2 2 2 ∂ ∂ ∂ ∂ ∂ ′ ′ Dx(f g)= f(x)g(x ) = 2 2 + 2 f(x)g(x ) · ∂x − ∂x′ ∂x − ∂x∂x′ ∂x′ ′ x′=x x =x

=f g 2f g + fg , (D.3) xx − x x xx . .

Note that Hirota’s operator Dx acts on a product of two functions in a similar was as Liebnitz rule for derivatives of a product but with alternating signs. More precisely, the signs of terms having an odd number of derivatives on the second function are negative.

Some useful properties include

∂nf Dn(f 1) = ,Dn(f f)=0, for n odd, (D.4) x · ∂xn x · Dn(f g)=( 1)nDn(g f), (D.5) x · − x · where n is a non-negative integer. The operator Dt is deﬁned similarly, ∂ ∂ Dt(f(t) g(t)) := f(t)g(t′) . (D.6) · ∂t − ∂t ′ ′ t =t

130 More generally, for multi-variable functions f(x,t) and g(x,t),

m n m n ∂ ∂ ∂ ∂ D D (f(x,t) g(x,t)) := f(x,t)g(x′,t′) , (D.7) x t ′ · ∂x − ∂x′ ∂t − ∂t′ x =x t′=t

for non-negative integers m and n.

Properties of the Hirota operators acting on multi-variable functions (some of which were used in Section 3.4) include

D D (f 1) = f = D D (1 f), (D.8) x t · xt x t · D4(f 1) = f = D4(1 f), (D.9) x · 4x x · DmDn(eη1 eη2 )=(K K )m(Ω Ω )neη1+η2 , (D.10) x t · 1 − 2 1 − 2

for ηi = Kix +Ωit + δi, i =1, 2, where Ki, Ωi,δi are constants, ∂ ∂ ∂ ∂ D D (f f)= f(x,t)f(x′,t′) x t ′ · ∂x − ∂x′ ∂t − ∂t′ x =x t′=t

= f f + ff f f f f xt xt − t x − x t = 2(ff f f ), (D.11) xt − x t 4 4 ∂ ∂ Dx(f f)= f(x,t)f(x′,t′) · ∂x − ∂x′ x′=x ′ t =t = f f 4f f +6f f 4f f + ff 4x − 3x x xx xx − x 3x 4x = 2(ff 4f f +3f 2 ). (D.12) 4x − 3x x xx

Adding (D.11) and (D.12) gives the bilinear form of the KdV equation shown in Section 3.4. Many more properties of the D-operators (also known as Hirota derivatives) can be found in [75, Appendix I] and [72, Section 1.6].

Hirota [72, p. 28] also deﬁned the operator Dz and diﬀerential operator ∂z as follows

Dz = Dt + εDx and ∂z = ∂t + ε∂x. (D.13)

Then, for f(x,t) and g(x,t) one has

131 Dn(f g)=(D + εD )n(f g) (D.14) z · t x · n n 1 n k n k k n n = D + nεD − D + + ε D − D + + ε D (f g) (D.15) t t x ··· k t x ··· x · n n k n k k = ε Dt − Dx (f g). (D.16) k ! · Xk=0 Using this formula one can calculate various products of powers of the Dt and Dx operators very fast. For example, from (D.16) we see that 3D D2(f g) is the coeﬃcient of ε2 in t x · D3(f g). z ·

An equivalent deﬁnition [72, p. 23] of the Dx and Dt operators is

m n m n ∂ ∂ Dx Dt (f(x,t) g(x,t)) = f(x + y,t + s)g(x y,t s) , (D.17) · ∂ym ∂sn − − y=0

s=0 for m,n = 0, 1, 2,.... Again, this is like Leibniz’ rule for derivatives but with alternating signs. For comparison, Leibniz’ rule for a product of two functions can be deﬁned as

m n m n ∂ ∂ Dx Dt (f(x,t)g(x,t)) := f(x + y,t + s)g(x + y,t + s) , (D.18) ∂ym ∂sn y=0

s=0

for m,n =0, 1, 2,....

Note that the usual Taylor series of a function f(x) can be formally expressed as

∞ δk f(x + δ)= ∂kf(x) = exp(δ∂ )f(x). (D.19) k! x x Xk=0 k By analogy, one can deﬁne the Hirota derivative, Dx, of order k with respect to x by

∞ yk f(x + y)g(x y)= Dk(f g) = exp(yD )(f(x) g(x)), (D.20) − k! x · x · k=0 X which is readily veriﬁed by taking Taylor expansions of f(x + y) and g(x y) and grouping − terms in powers in y.

In addition to the various deﬁnitions given above, Hirota’s Dx operator may also be deﬁned [72, p. 29] as

132 ∂ ∂ exp(δDx)(f(x) g(x)) := exp δ( ) (f(x)g(x′)) · ∂x − ∂x ′ ′ ! x =x

:= exp(δ∂ )f(x + y)g(x y) y − y=0

= f(x + δ)g(x δ), (D.21) −

where δ is a parameter. Similar formulas hold for Dt and Dz. To show how (D.2) and (D.3) follow from (D.21), introduce formal series expansions of exp(δD ),f(x + δ) and g(x δ), to replace (D.21) by x −

1 1 I + δD + δ2D2 + δ2D3 + ... (f(x) g(x)) x 2 x 6 x · 1 1 = f + δf + δ2f + δ3f + x 2 xx 6 xxx ··· × 1 1 g δg + δ2g δ3g + − x 2 xx − 6 xxx ··· 1 2 = fg + δ (fxg fgx)+ δ (fxxg 2fxgx + fgxx)+ , (D.22) − 2 − ··· !

and equate the coefficients of matching powers in δ on the left and right hand sides of (D.22). For example, (D.2) and (D.3) arise from equating the coefficients in δ and δ2, respectively. Definition (D.21) allows for a smooth transition from continuous to discrete bilinear operators.

D.2 Hirota’s Diﬀerence Operator

In (D.21), the parameter δ was assumed to be continuously changing. However, δ can also take on discrete values. If δ is assumed to be a lattice parameter, i.e., the distance between lattice points n and n + 1, then

∂ ∂ exp(δDn)(f(n) g(n)) := exp δ( ) (f(n)g(n′)) := f(n + δ)g(n δ), (D.23) · ∂n − ∂n ′ − ′ ! n =n

deﬁne the discrete analogue of (D.21), where Dn is Hirota’s bilinear diﬀerence operator.

133 Based on (D.23), one deﬁnes

1 cosh(δDn)= exp(δDn) + exp( δDn) , (D.24) 2 − ! 1 sinh(δDn)= exp(δDn) exp( δDn) , (D.25) 2 − − ! which both play an important role in Section 3.4.1.

In the same spirit, Hirota [5] deﬁned diﬀerence operators Dx and Dt as follows

exp(εD + δD )(a(x,t) b(x,t)) := a(x + ε,t + δ)b(x ε,t δ), (D.26) x t · − − where one can replace x by n, t by m, ε by p, and δ by q when dealing with nonlinear P∆Es. Similarly, one can deﬁne a discrete Hirota derivative for multi-variable functions as

f(x + y ,x + y ,...)g(x y ,x y ,...) 1 1 2 2 1 − 1 2 − 2 = ey1D1+y2D2+...(f(x ,x , ) g(x ,x , )), (D.27) 1 2 ··· · 1 2 ···

where D1 is the Hirota derivative with respect to x1, etc. For example,

f(x + y,t + s)g(x y,t s)= eyDx+sDt (f(x,t) g(x,t)). (D.28) − − · Identities and formulas involving various D-operators can be found in [5, Appendix I].

134