A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations

TL;DR

This paper introduces a four-parameter family of wave-breaking equations that generalize the Camassa-Holm and Novikov equations, analyzing their conservation laws, solutions, and symmetries, with a focus on a particular interesting case.

Contribution

It presents a new four-parameter polynomial family of equations extending known wave-breaking models and classifies their key properties, including conservation laws and solutions.

Findings

Identifies a 1-parameter equation with features similar to Camassa-Holm and Novikov equations.

Shows the existence of N-peakon solutions and conserved $H^1$ norm.

Demonstrates wave-breaking phenomena during peakon collisions.

Abstract

A 4-parameter polynomial family of equations generalizing the Camassa-Holm and Novikov equations that describe breaking waves is introduced. A classification of low-order conservation laws, peaked travelling wave solutions, and Lie symmetries is presented for this family. These classifications pick out a 1-parameter equation that has several interesting features: it reduces to the Camassa-Holm and Novikov equations when the polynomial has degree two and three; it has a conserved $H^1$ norm and it possesses $N$-peakon solutions, when the polynomial has any degree; and it exhibits wave-breaking for certain solutions describing collisions between peakons and anti-peakons in the case $N=2$.