Symbolic Computation of Lax Pairs of Partial Difference Equations Using Consistency Around the Cube

TL;DR

This paper reviews and extends a three-step method for deriving Lax pairs of scalar and system partial difference equations, including new Lax pairs for recently discovered equations, with an algorithmic implementation in Mathematica.

Contribution

It extends the Nijhoff and Bobenko & Suris method to systems of P extDelta Es and provides new Lax pairs for recent equations, enhancing the integrability analysis toolkit.

Findings

Derived Lax pairs for scalar integrable P extDelta Es.

Presented Lax pairs for systems including KdV, nonlinear Schrödinger, and Boussinesq lattice systems.

Implemented the method algorithmically in Mathematica.

Abstract

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (P\Delta Es) is reviewed. The method assumes that the P\Delta Es are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of P\Delta Es where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable P\Delta Es classified by Adler, Bobenko, and Suris and systems of P\Delta Es including the integrable 2-component potential Korteweg-de Vries lattice system, as well as nonlinear Schroedinger and Boussinesq-type lattice systems. Previously unknown Lax pairs are presented for P\Delta Es recently derived by Hietarinta (J. Phys. A: Math. Theor., 44, 2011, Art. No. 165204). The method is algorithmic and is being implemented in Mathematica.