On relationships between symmetries depending on arbitrary functions and integrals of discrete equations

TL;DR

This paper investigates the link between symmetries depending on arbitrary functions and integrability of discrete equations, proving a formal version of the conjecture for semi-discrete and discrete cases through Laplace invariants and formal integrals.

Contribution

It extends the conjecture relating symmetries and integrability to semi-discrete and discrete equations, using Laplace invariants and formal integrals.

Findings

Sequence of Laplace invariants terminates by zero for equations with symmetry operators.

Vanishing Laplace invariant enables construction of formal integrals.

Proves a formal version of the symmetry-integrability conjecture for discrete equations.

Abstract

The paper is devoted to the conjecture that an equation is Darboux integrable if and only if it possesses symmetries depending on arbitrary functions. We note that results of previous works together prove this conjecture for scalar partial differential equations of the form $u_{xy}=F(x,y,u,u_x,u_y)$. For autonomous semi-discrete and discrete analogues of these equations we prove that the sequence of Laplace invariants is terminated by zero for an equation if this equation admits an operator mapping any function of one independent variable into a symmetry of the equation. The vanishing of an Laplace invariant allows us to construct a formal integral, i.e. an operator that maps symmetries into integrals (including, generally speaking, trivial integrals). This and results of previous works together prove a `formal' version of the aforementioned conjecture in the semi-discrete and pure discrete cases.