A dispersive regularization for the modified Camassa-Holm equation

TL;DR

This paper introduces a dispersive regularization method for the modified Camassa-Holm equation, enabling the construction of global peakon solutions that do not collide, and extends to general initial data via a mean field limit.

Contribution

It develops a novel dispersive regularization approach for the mCH equation, producing collision-free peakon solutions and establishing a framework for weak solutions with general initial data.

Findings

Constructed collision-free N-peakon solutions.

Derived a system of ODEs for peakon dynamics.

Obtained global weak solutions for general initial data.

Abstract

In this paper, we present a dispersive regularization for the modified Camassa-Holm equation (mCH) in one dimension, which is achieved through a double mollification for the system of ODEs describing trajectories of $N$-peakon solutions. From this regularized system of ODEs, we obtain approximated $N$-peakon solutions with no collision between peakons. Then, a global $N$-peakon solution for the mCH equation is obtained, whose trajectories are global Lipschitz functions and do not cross each other. When $N=2$, the limiting solution is a sticky peakon weak solution. By a limiting process, we also derive a system of ODEs to describe $N$-peakon solutions. At last, using the $N$-peakon solutions and through a mean field limit process, we obtain global weak solutions for general initial data $m_0$ in Radon measure space.