The converse problem for the multipotentialisation of evolution equations and systems

TL;DR

This paper introduces a method to identify and classify evolution equations that can be multipotentialised into target equations, extending to higher dimensions and revealing connections to auto-Bäcklund transformations with several explicit examples.

Contribution

It presents a novel converse methodology for classifying multipotentialisable evolution equations and systems, including higher-dimensional cases and auto-Bäcklund transformations.

Findings

Identified classes of linearisable third-order evolution equations

Analyzed a fifth-order symmetry-integrable evolution equation

Derived connections between the converse problem and auto-Bäcklund transformations

Abstract

We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the {\it converse problem}. Although we mainly study a method for $(1+1)$-dimensional equations/system, we do also propose an extension of the methodology to higher-dimensional evolution equations. An important point is that the proposed converse method allows one to identify certain types of auto-B\"acklund transformations for the equations/systems. In this respect we define the {\it triangular-auto-B\"acklund transformation} and derive its connections to the converse problem. Several explicit examples are given. In particular we investigate a class of linearisable third-order evolution equations, a fifth-order symmetry-integrable evolution equation as well as linearisable systems.