On relationships between symmetries depending on arbitrary functions and integrals of 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.