Patched peakon weak solutions of the modified Camassa-Holm equation

TL;DR

This paper introduces a novel patching technique to construct bounded, single-valued peakon weak solutions for the modified Camassa-Holm equation, including solutions with jump conditions and applications to closed planar curve flows.

Contribution

The paper develops a new patching method to generate bounded peakon weak solutions of the mCH equation, extending the solution class and analyzing their geometric properties.

Findings

Constructed patched bounded peakon solutions satisfying jump conditions.

Derived solutions using the Helmholtz operator fundamental solution.

Analyzed length and area-preserving curve flows with peakon solutions.

Abstract

In this paper, we study traveling wave solutions and peakon weak solutions of the modified Camassa-Holm (mCH) equation with dispersive term $2ku_x$ for $k\in\mathbb{R}$. We study traveling wave solutions through a Hamiltonian system obtained from the mCH equation by using a nonlinear transformation. The typical traveling wave solutions given by this Hamiltonian system are unbounded or multi-valued. We provide a method, called patching technic, to truncate these traveling wave solutions and patch different segments to obtain patched bounded single-valued peakon weak solutions which satisfy jump conditions at peakons. Then, we study some special peakon weak solutions constructed by the fundamental solution of the Helmholtz operator $1-\partial_{xx}$, which can also be obtained by the patching technic. At last, we study some length and total signed area preserving closed planar curve flows that can be described by the mCH equation when $k=1$, for which we give a Hamiltonian structure and use the patched periodic peakon weak solutions to investigate loops with cusps.