On Darboux Integrable Semi-Discrete Chains

TL;DR

This paper investigates Darboux integrable semi-discrete chains, establishing canonical forms for their integrals and providing methods to explicitly construct general solutions for certain classes.

Contribution

It introduces a canonical form for integrals of Darboux integrable chains and develops a method to explicitly construct their general solutions.

Findings

Integrals can be transformed into a canonical form.

Explicit formulas for solutions are derived for specific classes.

A new method for constructing solutions to Darboux integrable chains.

Abstract

Differential-difference equation $\frac{d}{dx}t(n+1,x)=f(x,t(n,x),t(n+1,x),\frac{d}{dx}t(n,x))$ with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$ is studied. We call an equation of such kind Darboux integrable, if there exist two functions (called integrals) $F$ and $I$ of a finite number of dynamical variables 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: $Dp(n)=p(n+1)$. It is proved that the integrals can be brought to some canonical form. A method of construction of an explicit formula for general solution to Darboux integrable chains is discussed and for a class of chains such solutions are found.