Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation

TL;DR

This paper investigates the Hamiltonian structure of peakon solutions for the modified Camassa-Holm equation, revealing explicit conserved integrals and differences in Hamiltonian properties between even and odd numbers of peakons.

Contribution

It provides an explicit conserved integral for all peakon numbers and clarifies the Hamiltonian structure differences between even and odd peakon solutions.

Findings

Explicit conserved integral for N-peakon solutions with N ≥ 2.

Hamiltonian structure valid for even N using a natural Poisson bracket.

Loss of conservation for the original Hamiltonians in peakon solutions.

Abstract

The modified Camassa-Holm (mCH) equation is a bi-Hamiltonian system possessing $N$-peakon weak solutions, for all $N\geq 1$, in the setting of an integral formulation which is used in analysis for studying local well-posedness, global existence, and wave breaking for non-peakon solutions. Unlike the original Camassa-Holm equation, the two Hamiltonians of the mCH equation do not reduce to conserved integrals (constants of motion) for $2$-peakon weak solutions. This perplexing situation is addressed here by finding an explicit conserved integral for $N$-peakon weak solutions for all $N\geq 2$. When $N$ is even, the conserved integral is shown to provide a Hamiltonian structure with the use of a natural Poisson bracket that arises from reduction of one of the Hamiltonian structures of the mCH equation. But when $N$ is odd, the Hamiltonian equations of motion arising from the conserved integral using this Poisson bracket are found to differ from the dynamical equations for the mCH $N$-peakon weak solutions. Moreover, the lack of conservation of the two Hamiltonians of the mCH equation when they are reduced to $2$-peakon weak solutions is shown to extend to $N$-peakon weak solutions for all $N\geq 2$. The connection between this loss of integrability structure and related work by Chang and Szmigielski on the Lax pair for the mCH equation is discussed.