A New Integrable Equation with Peakon Solutions

TL;DR

This paper introduces a new integrable partial differential equation similar to the Camassa-Holm equation, proves its integrability through a Lax pair, and explores its peakon solutions and related dynamics.

Contribution

It presents a novel integrable equation, establishes its Lax pair, and connects it to the Kaup-Kupershmidt hierarchy, expanding the understanding of peakon solutions.

Findings

The new equation is exactly integrable.

It admits multi-peakon solutions.

Finite-dimensional peakon dynamics are characterized.

Abstract

We consider a new partial differential equation, of a similar form to the Camassa-Holm shallow water wave equation, which was recently obtained by Degasperis and Procesi using the method of asymptotic integrability. We prove the exact integrability of the new equation by constructing its Lax pair, and we explain its connection with a negative flow in the Kaup-Kupershmidt hierarchy via a reciprocal transformation. The infinite sequence of conserved quantities is derived together with a proposed bi-Hamiltonian structure. The equation admits exact solutions in the form of a superposition of multi-peakons, and we describe the integrable finite-dimensional peakon dynamics and compare it with the analogous results for Camassa-Holm peakons.