Elastic collisions among peakon solutions for the Camassa–Holm equation

Abstract

The purpose of this paper is to study the dynamics of the interaction among a special class of solutions of the one-dimensional Camassa-Holm equation. The equation yields soliton solutions whose identity is preserved through nonlinear interactions. These solutions are characterized by a discontinuity at the peak in the wave shape and are thus called peakon solutions. We apply a particle method to the Camassa-Holm equation and show that the nonlinear interaction among the peakon solutions resembles an elastic collision, i.e., the total energy of the system before the peakon interaction is equal to the total energy and momentum of the system after the collision. From this result, we provide several numerical illustrations which supports the analytical study, as well as showcase the merits of using a particle method to simulate solutions to the Camassa-Holm equation under a wide class of initial data.

Bäcklund Transformations for the Camassa–Holm Equation

“…This system for s is equivalent to the Lax pair for CH. Note (12) can be simplified with the help of (11) and (1) to…”

“…There is an inverse scattering formalism [13,14], explicit formulas can be found for multipeakon,multisoliton, multicuspon and soliton-cuspon solutions [60,4,5,6,22,31,39,49,18,38,43,53,44,51,52,19,63] there are an infinite number of local conservation laws [23,56,57,28,25,27,35,11,30,24], and there is a rich algebra of symmetries [56,57,28,25,27,24]. Other significant works on CH include studies of the stability of peakon and other exact solutions [15,16,17,36] and interesting numerical studies [32,45,21,12].…”

Global Convergence of a Sticky Particle Method for the Modified Camassa--Holm Equation

“…The peakons in the CH equation elastically bounce back after becoming close to each other, and they exchange momentum. Hence, the total energy p 2 1 + p 2 2 is conserved [10]. However, two peakons for the mCH equation can collide in finite time and for the sticky collision, the energy becomes…”

A Particle Method for the b-Equation

“…In this section, we consider the b-equation (1). As mentioned above, Equation (1) coincides with the dispersionless C-H equation in [3] and D-P equation in [4]. It was first derived by using asymptotic expansions directly in the Hamiltonian for Euler's equations in the shallow water regime.…”