Complete list of Darboux Integrable Chains of the form $t_{1x}=t_x+d(t,t_1)$

TL;DR

This paper classifies Darboux integrable differential-difference equations of a specific form, providing a complete list when the function involved has a particular structure, using Lie algebra methods.

Contribution

The paper offers a complete classification of Darboux integrable chains of a certain form, utilizing characteristic Lie algebra techniques for the first time in this context.

Findings

Complete list of Darboux integrable chains with specific function form

Effective classification method using Lie algebra finiteness

Identification of integrable equations within the specified class

Abstract

We study differential-difference equation of the form $$ \frac{d}{dx}t(n+1,x)=f(t(n,x),t(n+1,x),\frac{d}{dx}t(n,x)) $$ with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$. Equation of such kind is called Darboux integrable, if there exist two functions $F$ and $I$ of a finite number of arguments $x$, $\{t(n\pm k,x)\}_{k=-\infty}^\infty$, ${\frac{d^k}{dx^k}t(n,x)}_{k=1}^\infty$, 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: $Dp(n)=p(n+1)$. Reformulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations is given in the case when the function $f$ is of the special form $f(u,v,w)=w+g(u,v)$.