Peakon-antipeakon interactions in the Degasperis-Procesi Equation

TL;DR

This paper develops a formalism to analyze peakon-antipeakon collisions in the Degasperis-Procesi equation, revealing collision rules, shock formation, and spectral properties linked to initial configurations.

Contribution

It introduces a general framework for studying peakon collisions in the DP equation and explores spectral correlations with initial peakon-antipeakon arrangements.

Findings

Peakons collide only in pairs, no triple collisions occur.

A shockpeakon is formed at each collision.

Spectral properties depend on initial configurations, with real spectra under certain conditions.

Abstract

Peakons are singular, soliton-like solutions to nonlinear wave equations whose dynamics can be studied using ordinary differential equations (ODEs). The Degasperis-Procesi equation (DP) is an important example of an integrable PDE exhibiting wave breaking in the peakon sector thus affording an interpretation of wave breaking as a mechanical collision of particles. In this paper we set up a general formalism in which to study collisions of DP peakons and apply it, as an illustration, to a detailed study of three colliding peakons. It is shown that peakons can collide only in pairs, no triple collisions are allowed and at the collision a shockpeakon is created. We also show that the initial configuration of peakon-antipeakon pairs is nontrivially correlated with the spectral properties of an accompanying non-selfadjoint boundary value problem. In particular if peakons or antipeakons are bunched up on one side relative to a remaining antipeakon or peakon then the spectrum is real and simple. Even though the spectrum is in general complex the existence of a global solution in either time direction dynamics is shown to imply the reality of the spectrum of the boundary value problem.