Partial Lie-point symmetries of differential equations

TL;DR

This paper introduces the concept of partial Lie-point symmetries for differential equations, allowing the identification of solution subsets invariant under specific transformations, extending traditional symmetry analysis.

Contribution

It defines partial symmetries and provides an effective procedure to determine them and their invariant solution subsets, including extensions to generalized and discrete symmetries.

Findings

Effective method for finding partial symmetries demonstrated with examples

Partial symmetries relate to conditional symmetries and dynamical systems

Procedure extends to generalized and discrete symmetries

Abstract

When we consider a differential equation $\Delta=0$ whose set of solutions is ${{\cal S}}_\Delta$, a Lie-point exact symmetry of this is a Lie-point invertible transformation $T$ such that $T({{\cal S}}_\Delta)={{\cal S}}_\Delta$, i.e. such that any solution to $\Delta=0$ is tranformed into a (generally, different) solution to the same equation; here we define {\it partial} symmetries of $\Delta=0$ as Lie-point invertible transformations $T$ such that there is a nonempty subset ${{\cal P}} \subset {{\cal S}}_\Delta$ such that $T({{\cal P}}) = {{\cal P}}$, i.e. such that there is a subset of solutions to $\Delta=0$ which are transformed one into the other. We discuss how to determine both partial symmetries and the invariant set ${{\cal P}} \subset {{\cal S}}_\Delta$, and show that our procedure is effective by means of concrete examples. We also discuss relations with conditional symmetries, and how our discussion applies to the special case of dynamical systems. Our discussion will focus on continuous Lie-point partial symmetries, but our approach would also be suitable for more general classes of transformations; the discussion is indeed extended to partial generalized (or Lie-B\"acklund) symmetries along the same lines, and in the appendix we will discuss the case of discrete partial symmetries.