Symbolic Computation of Recursion Operators for Nonlinear Differential-Difference equations

TL;DR

This paper introduces an algorithm for symbolically computing recursion operators in nonlinear differential-difference equations, which are crucial for establishing integrability and generating symmetries, implemented in Mathematica.

Contribution

The paper presents a novel algorithm that automates the symbolic computation of recursion operators for DDEs, extending previous methods and applied to well-known integrable systems.

Findings

Successfully computed recursion operators for Kac-van Moerbeke, Toda, and Ablowitz-Ladik lattices.

Demonstrated the algorithm's effectiveness in establishing integrability.

Implemented the algorithm in a Mathematica package for broader use.

Abstract

An algorithm for the symbolic computation of recursion operators for systems of nonlinear differential-difference equations (DDEs) is presented. Recursion operators allow one to generate an infinite sequence of generalized symmetries. The existence of a recursion operator therefore guarantees the complete integrability of the DDE. The algo-rithm is based in part on the concept of dilation invariance and uses our earlier algorithms for the symbolic computation of conservation laws and generalized symmetries. The algorithm has been applied to a number of well-known DDEs, including the Kac-van Moerbeke (Volterra), Toda, and Ablowitz-Ladik lattices, for which recursion opera-tors are shown. The algorithm has been implemented in Mathematica, a leading com-puter algebra system. The package DDERecursionOperator.m is briefly discussed.