Symmetries and reductions of order for certain nonlinear third and second-order differential equations with arbitrary nonlinearity

TL;DR

This paper explores methods to reduce the order of specific nonlinear second- and third-order differential equations using symmetries and transformations, including a folding transformation that simplifies arbitrary nonlinear exponents.

Contribution

It introduces a folding transformation technique to convert arbitrary nonlinear exponents into fixed numerical exponents, facilitating order reduction of nonlinear differential equations.

Findings

Order reduction achieved through symmetry analysis

Folding transformation simplifies arbitrary nonlinearities

Applicable to certain classes of nonlinear differential equations

Abstract

We examine the reductions of the order of certain third- and second-order nonlinear equations with arbitrary nonlinearity through their symmetries and some appropriate transformations. We use the folding transformation which enables one to change from a nonlinearity with an arbitrary exponent to a nonlinearity with a specific numerical exponent.