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 non- linear 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 poten- tial 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 trans- formations 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 min- imal 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.