On existence of an x-integral for a semi-discrete chain of hyperbolic type

TL;DR

This paper investigates semi-discrete hyperbolic chains and establishes verifiable conditions for the existence of an x-integral of minimal order 4, contributing to the understanding of integrability in such systems.

Contribution

It provides explicit, easily checkable criteria for the existence of minimal order 4 x-integrals in semi-discrete hyperbolic chains.

Findings

Conditions for x-integral existence are derived.

Criteria are verifiable and applicable to specific chains.

Enhances understanding of integrability in semi-discrete hyperbolic systems.

Abstract

A class of semi-discrete chains of the form $t_{1x} = f(x, t, t_1, t_x)$ is considered. For the given chains easily verifiable conditions for existence of x-integral of minimal order 4 are obtained.