Necessary integrability conditions for evolutionary lattice equations

TL;DR

This paper investigates the solution structure of Lax equations for formal power series in the context of evolutionary lattice equations, deriving necessary conditions for their integrability.

Contribution

It introduces a new property linking solutions of different degrees, leading to necessary integrability conditions for scalar evolutionary lattices.

Findings

Existence of a solution H of degree m such that H^k=G^m when a solution G of degree k exists.

Derived necessary integrability conditions for scalar evolutionary lattice equations.

Established a structural property of solutions to the Lax equation for formal power series.

Abstract

The structure of solutions is studied for the Lax equation $D_t(G)=[F,G]$ for formal power series with respect to the shift operator. It is proved that if the equation with a given series $F$ of degree $m$ admits a solution $G$ of degree $k$ then it admits, as well, a solution $H$ of degree $m$ such that $H^k=G^m$. This property is used for derivation of necessary integrability conditions for scalar evolutionary lattices.