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Symbolic computation of Lax pairs of nonlinear partial difference equations
Bridgman, Terry J.
Bridgman, Terry J.
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2018
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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.
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