A general family of multi-peakon equations and their properties

TL;DR

This paper introduces a broad family of peakon equations, demonstrating their weak solutions, wave breaking criteria, and Hamiltonian structures, unifying known equations and revealing new behaviors of multi-peakon solutions.

Contribution

It generalizes peakon equations to include all known breaking wave equations, providing new insights into their solutions, structures, and behaviors without requiring integrability or Hamiltonian frameworks.

Findings

All equations in the family have multi-peakon weak solutions.

Existence of single and generalized peakon solutions under simple conditions.

Novel behaviors of 2-peakon solutions, including peakon-antipeakon bound states.

Abstract

A general family of peakon equations is introduced, involving two arbitrary functions of the wave amplitude and the wave gradient. This family contains all of the known breaking wave equations, including the integrable ones: Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation, and FORQ/modified Camassa-Holm equation. One main result is to show that all of the equations in the general family possess weak solutions given by multi-peakons which are a linear superposition of peakons with time-dependent amplitudes and positions. In particular, neither an integrability structure nor a Hamiltonian structure is needed to derive $N$-peakon weak solutions for arbitrary $N>1$. As a further result, single peakon travelling-wave solutions are shown to exist under a simple condition on one of the two arbitrary functions in the general family of equations, and when this condition fails, generalized single peakon solutions that have a time-dependent amplitude and a time-dependent speed are shown to exist. Wave breaking criteria are outlined for this general family of peakon equations. An interesting generalization of the Camassa-Holm and FORQ/modified Camassa-Holm equations is obtained by deriving the most general subfamily of peakon equations that possess the Hamiltonian structure shared by the Camassa-Holm and FORQ/modified Camassa-Holm equations. Peakon travelling-wave solutions and their features, including a variational formulation (minimizer problem), are derived for these generalized equations. A final main result is that $2$-peakon weak solutions are investigated and shown to exhibit several novel kinds of behaviour, including the formation of a bound pair consisting of a peakon and an anti-peakon that have a maximum finite separation.