Liouville integrability of conservative peakons for a modified CH equation

TL;DR

This paper demonstrates that the conservative peakon equations of the modified Camassa-Holm equation are Liouville integrable by establishing a suitable Hamiltonian structure and applying inverse spectral methods.

Contribution

It introduces a Poisson structure making the conservative peakon equations Hamiltonian and proves their Liouville integrability using inverse spectral techniques.

Findings

Peakons preserve the Sobolev H^1 norm in the conservative sector.

The equations are shown to be Hamiltonian with an appropriate Poisson structure.

Liouville integrability is established through inverse spectral analysis.

Abstract

The modified Camassa-Holm equation (also called FORQ) is one of numerous $cousins$ of the Camassa-Holm equation possessing non-smoth solitons ($peakons$) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissapative) the Sobolev $H^1$ norm is not preserved, in the other sector (conservative), introduced in [2], the time evolution of peakons leaves the $H^1$ norm invariant. In this Letter, it is shown that the conservative peakon equations of the modified Camassa-Holm can be given an appropriate Poisson structure relative to which the equations are Hamiltonian and, in fact, Liouville integrable. The latter is proved directly by exploiting the inverse spectral techniques, especially asymptotic analysis of solutions, developed elsewhere (in [3]).