Equivalence transformations in the study of integrability

TL;DR

This paper explores how point transformations and equivalence groupoids can be used to identify and derive integrable variable-coefficient differential equations, with a focus on evolution equations like the KdV class.

Contribution

It introduces a systematic method for using equivalence transformations to analyze and classify integrable variable-coefficient differential equations, especially evolution equations.

Findings

Derived classes of integrable variable-coefficient equations

Established the structure of the equivalence groupoid for these classes

Applied the method to fifth-order KdV-like equations

Abstract

We discuss how point transformations can be used for the study of integrability, in particular, for deriving classes of integrable variable-coefficient differential equations. The procedure of finding the equivalence groupoid of a class of differential equations is described and then specified for the case of evolution equations. A class of fifth-order variable-coefficient KdV-like equations is studied within the framework suggested.