On the Classification of Darboux Integrable Chains

TL;DR

This paper classifies Darboux integrable differential-difference equations of a specific form using characteristic Lie algebras, providing a complete list of equations with nontrivial x-integrals when f has a particular structure.

Contribution

It introduces an effective classification method for Darboux integrable chains based on characteristic Lie algebras and provides a complete classification for a special class of functions f.

Findings

Complete list of Darboux integrable chains with specific f form

Effective classification method using characteristic Lie algebras

Identification of equations admitting nontrivial x-integrals

Abstract

We study differential-difference equation of the form $t_{x}(n+1)=f(t(n),t(n+1),t_x(n))$ with unknown $t=t(n,x)$ depending on $x$, $n$. The equation is called Darboux integrable, if there exist functions $F$ (called an $x$-integral) and $I$ (called an $n$-integral), both of a finite number of variables $x$, $t(n)$, $t(n\pm 1)$, $t(n\pm 2)$, $...$, $t_x(n)$, $t_{xx}(n)$, $...$, such that $D_xF=0$ and $DI=I$, where $D_x$ is the operator of total differentiation with respect to $x$, and $D$ is the shift operator: $Dp(n)=p(n+1)$. The Darboux integrability property is reformulated in terms of characteristic Lie algebras that gives an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial $x$-integrals is given in the case when the function $f$ is of the special form $f(x,y,z)=z+d(x,y)$.