Singular reduction modules of differential equations

TL;DR

This paper introduces the concept of singular reduction modules for differential equations, showing how they can improve the derivation of nonclassical symmetries and facilitate reductions to simpler forms.

Contribution

It defines singular reduction modules, analyzes their properties, and relates them to order reduction and previous no-go results in nonclassical symmetry methods.

Findings

Singular modules can lower the order of reduced equations.

Differential equations with parameterized singular modules are characterized.

Reductions to algebraic and first-order ODEs are systematically studied.

Abstract

The notion of singular reduction modules, i.e., of singular modules of nonclassical (conditional) symmetry, of differential equations is introduced. It is shown that the derivation of nonclassical symmetries for differential equations can be improved by an in-depth prior study of the associated singular modules of vector fields. The form of differential functions and differential equations possessing parameterized families of singular modules is described up to point transformations. Singular cases of finding reduction modules are related to lowering the order of the corresponding reduced equations. As examples, singular reduction modules of evolution equations and second-order quasi-linear equations are studied. Reductions of differential equations to algebraic equations and to first-order ordinary differential equations are considered in detail within the framework proposed and are related to previous no-go results on nonclassical symmetries.