A precise definition of reduction of partial differential equations

TL;DR

This paper provides a precise definition of reduction in partial differential equations, unifies classical and non-classical symmetry methods, and demonstrates their equivalence through an example involving the nonlinear wave equation.

Contribution

It introduces a comprehensive definition of reduction, establishing the equivalence of non-classical and direct approaches in PDE reduction.

Findings

Established the equivalence of non-classical and direct reduction methods.

Provided a non-trivial example of non-classical reduction of a nonlinear wave equation.

Demonstrated the generalization of symmetry reductions beyond classical Lie methods.

Abstract

We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry (classical and non-classical) reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical (conditional symmetry) and direct (Ansatz) approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in (1+3) dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalization of the well-known symmetry reductions of the nonlinear wave equations.