Integrability test for evolutionary lattice equations of higher order

TL;DR

The paper introduces a generalized summation by parts algorithm to test the integrability of higher-order evolutionary lattice equations by analyzing their difference equations and formal symmetries.

Contribution

It presents a new algorithm for integrability testing of differential-difference equations, implemented in Mathematica, based on solving specific difference equations.

Findings

Algorithm effectively tests integrability of higher-order lattice equations.

Implementation in Mathematica facilitates practical application.

Provides a systematic approach for analyzing formal symmetries.

Abstract

A generalized summation by parts algorithm is presented for solving of difference equations of the form $T^m(y)-a[u]y=b[u]$ where $T$ denotes the shift $u_j\to u_{j+1}$. Solvability of such type of equations with respect to coefficients of formal symmetry (or formal recursion operator) provides a convenient integrability test for evolutionary differential-difference equations $u_{,t}=f(u_{-m},\dots,u_m)$. The algorithm is implemented in {\em Mathematica}.