Symbolic Computation of Conservation Laws, Generalized Symmetries, and Recursion Operators for Nonlinear Differential-Difference Equations

TL;DR

This paper presents algorithms for symbolically computing conservation laws, symmetries, and recursion operators for nonlinear differential-difference equations, aiding in testing their integrability.

Contribution

The authors develop and implement algorithms in Mathematica for the symbolic analysis of nonlinear DDEs, including conservation laws, symmetries, and recursion operators, demonstrated on the Toda lattice.

Findings

Algorithms successfully compute properties of nonlinear DDEs.

Implementation in Mathematica facilitates research on integrability.

The approach aids in identifying integrable systems.

Abstract

Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations (DDEs) are presented. The algorithms can be used to test the complete integrability of nonlinear DDEs. The ubiquitous Toda lattice illustrates the steps of the algorithms, which have been implemented in {\em Mathematica}. The codes {\sc InvariantsSymmetries.m} and {\sc DDERecursionOperator.m} can aid researchers interested in properties of nonlinear DDEs.