Unfamiliar Aspects of B\"acklund Transformations and an Associated Degasperis-Procesi Equation

TL;DR

This paper explores Bäcklund transformations for an associated Degasperis-Procesi equation, revealing their connection to symmetries and generating new solutions, thus advancing understanding of integrable systems.

Contribution

It introduces a new Bäcklund transformation for the aDP equation, along with its superposition principle and the method to derive infinitesimal symmetries.

Findings

Derived a Bäcklund transformation for the aDP equation

Established the superposition principle for the BT

Generated solutions and symmetries of the aDP equation

Abstract

We summarize the results of our recent work on B\"acklund transformations (BTs), particularly focusing on the relationship of BTs and infinitesimal symmetries. We present a BT for an associated Degasperis-Procesi (aDP) equation and its superposition principle, and investigate the solutions generated by application of this BT. Following our general methodology, we use the superposition principle of the BT to generate the infinitesimal symmetries of the aDP equation.