On Partial Differential and Difference Equations with Symmetries Depending on Arbitrary Functions

TL;DR

This paper explores conditions under which Lie symmetries of differential and difference equations depend on arbitrary functions, demonstrating their role as master symmetries through examples in both continuous and discrete settings.

Contribution

It provides insights into when symmetries depend on arbitrary functions and highlights their role as master symmetries in differential and difference equations.

Findings

Symmetries can depend on arbitrary functions in certain cases.

Generalized symmetries with arbitrary functions act as master symmetries.

Examples illustrate the dependence of symmetries on arbitrary functions.

Abstract

In this note we present some ideas on when Lie symmetries, both point and generalized, can depend on arbitrary functions. We show on a few examples, both in partial differential and partial difference equations when this happens. Moreover we show that the infinitesimal generators of generalized symmetries depending on arbitrary functions, both for continuous and discrete equations, effectively play the role of master symmetries.