A general family of multi-peakon equations and their properties

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.

36 pages; new material added on properties of the generalized CH and mCH/FORQ equations, including 2-peakon interactions

References in corpus (4)

• Degasperis-Procesi peakons and the discrete cubic string
• A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations
• Lax integrability and the peakon problem for the modified Camassa-Holm equation
• Complex oscillations in the delayed Fitzhugh-Nagumo equation

Cited by in corpus (5)

• Data-driven peakon and periodic peakon travelling wave solutions of some nonlinear dispersive equations via deep learning
• A formula for symmetry recursion operators from non-variational symmetries of partial differential equations
• Orbital stability of peakon solutions for a generalized higher-order Camassa-Holm equation
• The Cauchy problem and wave-breaking phenomenon for a generalized sine-type FORQ/mCH equation
• On the well-posedness of the Cauchy problem for the two-component peakon system in $C^k\cap W^{k,1}$