Degasperis-Procesi peakon dynamical system and finite Toda lattice of CKP type

TL;DR

This paper introduces a new finite Toda lattice of CKP type (C-Toda), explores its relation to the Degasperis-Procesi peakon system, and demonstrates their opposite flow dynamics through a Lax pair and continuum limit analysis.

Contribution

It proposes the finite C-Toda lattice with a Lax pair and establishes its connection to the DP peakon system as opposite flows, extending integrable systems theory.

Findings

Derived the finite C-Toda lattice and its Lax pair.

Established the DP peakon and C-Toda lattices as opposite flows.

Connected the continuum limit to a discrete CKP equation.

Abstract

In this paper, we propose a finite Toda lattice of CKP type (C-Toda) together with a Lax pair. Our motivation is based on the fact that the Camassa-Holm (CH) peakon dynamical system and the finite Toda lattice may be regarded as opposite flows in some sense. As an intriguing analogue to the CH equation, the Degasperis-Procesi (DP) equation also supports the presence of peakon solutions. Noticing that the peakon solution to the DP equation is expressed in terms of bimoment determinants related to the Cauchy kernel, we impose opposite time evolution on the moments and derive the corresponding bilinear equation. The corresponding quartic representation is shown to be a continuum limit of a discrete CKP equation, due to which we call the obtained equation finite Toda lattice of CKP type. Then, a nonlinear version of the C-Toda lattice together with a Lax pair is derived. As a result, it is shown that the DP peakon lattice and the finite C-Toda lattice form opposite flows under certain transformation.