Singular reduction operators in two dimensions

TL;DR

This paper introduces the concept of singular reduction operators for 2D PDEs, exhaustively describes their reductions to first-order ODEs, and demonstrates how to improve nonclassical symmetry derivations through analysis of singular vector fields.

Contribution

It defines singular reduction operators in two dimensions and systematically describes their reductions, providing new insights into nonclassical symmetries of evolution and wave equations.

Findings

Complete classification of reductions to first-order ODEs

Enhanced methods for deriving nonclassical symmetries

Analysis of singular operators in evolution and wave equations

Abstract

The notion of singular reduction operators, i.e., of singular operators of nonclassical (conditional) symmetry, of partial differential equations in two independent variables is introduced. All possible reductions of these equations to first-order ODEs are are exhaustively described. As examples, properties of singular reduction operators of (1+1)-dimensional evolution and wave equations are studied. It is shown how to favourably enhance the derivation of nonclassical symmetries for this class by an in-depth prior study of the corresponding singular vector fields.