Nonlocal invariance of the multipotentialisations of the Kupershmidt equation and its higher-order hierarchies

TL;DR

This paper explores the multipotentialisation process of the Kupershmidt equation, revealing new nonlocal invariance transformations and providing explicit recursion operators for its hierarchies, advancing understanding of nonlocal symmetries in integrable systems.

Contribution

It introduces new nonlocal invariance transformations derived from multipotentialisation of the Kupershmidt equation and its hierarchies, with explicit recursion operators provided.

Findings

New nonlocal invariance transformations identified.

Explicit recursion operators for hierarchies provided.

Enhanced understanding of multipotentialisation in integrable systems.

Abstract

The term multipotentialisation of evolution equations in $1+1$ dimensions refers to the process of potentialising a given evolution equation, followed by at least one further potentialisation of the resulting potential equation. For certain equations this process can be applied several times to result in a finite chain of potential equations, where each equation in the chain is a potential equation of the previous equation. By a potentialisation of an equation with dependent variable $u$ to an equation with dependent variable $v$, we mean a differential substitution $v_x=\Phi^t$, where $\Phi^t$ is a conserved current of the equation in $u$. The process of multipotentialisation may lead to interesting nonlocal transformations between the equations. Remarkably, this can, in some cases, result in nonlocal invariance transformations for the equations, which then serve as iteration formulas by which solutions can be generated for all the equations in the chain. In the current paper we give a comprehensive introduction to this subject and report new nonlocal invariance transformations that result from the multipotentialisation of the Kupershmidt equation and its higher-order hierarchies. The recursion operators that define the hierarchies are given explicitly.