On conservative sticky peakons to the modified Camassa-Holm equation

TL;DR

This paper proves the existence and uniqueness of conservative sticky peakon solutions to the modified Camassa-Holm equation using a sticky particle method and dispersion regularization, and explores non-uniqueness when peakon splitting is permitted.

Contribution

It introduces a dispersion regularization as a selection principle for unique conservative peakon solutions and demonstrates the sticky particle method's effectiveness for the modified Camassa-Holm equation.

Findings

Global existence of conservative sticky N-peakon solutions established.

Dispersion regularization ensures uniqueness of solutions.

Numerical results confirm the dispersion limit corresponds to sticky peakons.

Abstract

We use a sticky particle method to show global existence of (energy) conservative sticky $N$-peakon solutions to the modified Camassa-Holm equation. A dispersion regularization is provided as a selection principle for the uniqueness of conservative $N$-peakon solutions. The dispersion limit avoids the collision between peakons, and numerical results show that the dispersion limit is exactly the sticky peakons. At last, when the splitting of peakons is allowed, we give an example to show the non-uniqueness of conservative solutions.