Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation

TL;DR

This paper investigates singular peakon and compacton solutions in a non-integrable cross-coupled Camassa-Holm system, analyzing their unique 'waltzing' dynamics, collisions, and internal degrees of freedom through analytical and numerical methods.

Contribution

It introduces and analyzes novel peakon and compacton couple solutions in a non-integrable cross-coupled Camassa-Holm system, highlighting their complex inelastic collision behaviors and internal dynamics.

Findings

Peakon solutions exhibit 'waltzing' behavior with internal 'tempo' dynamics.

Collisions between peakon couples are inelastic and involve phase space cycling.

Compacton couples display similar collision phenomena with different shapes.

Abstract

We consider singular solutions of a system of two cross-coupled Camassa-Holm (CCCH) equations. This CCCH system admits peakon solutions, but it is not in the two-component CH integrable hierarchy. The system is a pair of coupled Hamiltonian partial differential equations for two types of solutions on the real line, each of which separately possesses exp(-|x|) peakon solutions with a discontinuity in the first derivative at the peak. However, there are no self-interactions, so each of the two types of peakon solutions moves only under the induced velocity of the other type. We analyse the `waltzing' solution behaviour of the cases with a single bound peakon pair (a peakon couple), as well as the over-taking collisions of peakon couples and the antisymmetric case of the head-on collision of a peakon couple and a peakon anti-couple. We then present numerical solutions of these collisions, which are inelastic because the waltzing peakon couples each possess an internal degree of freedom corresponding to their `tempo' -- that is, the period at which the two peakons of opposite type in the couple cycle around each other in phase space. Finally, we discuss compacton couple solutions of the cross-coupled Euler-Poincar\'e (CCEP) equations and illustrate the same types of collisions as for peakon couples, with triangular and parabolic compacton couples. We finish with a number of outstanding questions and challenges remaining for understanding couple dynamics of the CCCH and CCEP equations.