Generalized conditional symmetries of evolution equations

TL;DR

This paper explores the theoretical framework connecting generalized conditional symmetries of evolution equations with compatibility conditions, invariance, and solution families, providing new insights and a no-go theorem in the field.

Contribution

It establishes a formal relationship between generalized conditional symmetries, compatibility, and solution parametrization, including a no-go theorem on determining equations.

Findings

Revisits the connection between invariance and reduction of evolution equations.

Proves a no-go theorem on determining equations for symmetry operators.

Shows a one-to-one correspondence between symmetries and solution families.

Abstract

We analyze the relationship of generalized conditional symmetries of evolution equations to the formal compatibility and passivity of systems of differential equations as well as to systems of vector fields in involution. Earlier results on the connection between generalized conditional invariance and generalized reduction of evolution equations are revisited. This leads to a no-go theorem on determining equations for operators of generalized conditional symmetry. It is also shown that up to certain equivalences there exists a one-to-one correspondence between generalized conditional symmetries of an evolution equation and parametric families of its solutions.