Invariance of the Kaup-Kupershmidt equation and triangular auto-B\"acklund transformations

TL;DR

This paper introduces triangular auto-Bäcklund transformations for the Kaup-Kupershmidt equation, enabling solution generation through systematic multipotentialisation, and explores their invariance properties.

Contribution

It presents a novel systematic derivation of triangular auto-Bäcklund transformations for the Kaup-Kupershmidt equation, expanding solution-generation methods.

Findings

Auto-Bäcklund transformations enable new solutions.

Transformations derived from multipotentialisation.

Invariance properties of the Kaup-Kupershmidt equation analyzed.

Abstract

We report triangular auto-B\"acklund transformations for the solutions of a fifth-order evolution equation, which is a constraint for an invariance condition of the Kaup-Kupershmidt equation derived by E. G. Reyes in his paper titled "Nonlocal symmetries and the Kaup-Kupershmidt equation" [{\it J. Math. Phys.} {\bf 46}, 073507, 19 pp., 2005]. These auto-B\"acklund transformations can then be applied to generate solutions of the Kaup-Kupershmidt equation. We show that triangular auto-B\"acklund transformations result from a systematic multipotentialisation of the Kupershmidt equation.