Symbolic computation of Lax pairs of nonlinear partial difference equations

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. 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.