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Liouville integrability of conservative peakons for a modified CH equation

Xiang-Ke Chang Jacek Szmigielski

Dynamical Systems mathscidoc:1912.43172

Journal of nonlinear mathematical physics, 24, (4), 584-595, 2017.10
The modified Camassa-Holm (also called FORQ) equation is one of numerous <i>cousins</i> of the Camassa-Holm equation possessing non-smoth solitons (<i>peakons</i>) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissipative<sup>a</sup>) the Sobolev <i>H</i><sup>1</sup> norm is not preserved, in the other sector (conservative), introduced in [2], the time evolution of peakons leaves the <i>H</i><sup>1</sup> 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 [3].
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@inproceedings{xiang-ke2017liouville,
  title={Liouville integrability of conservative peakons for a modified CH equation},
  author={Xiang-Ke Chang, and Jacek Szmigielski},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112726491645732},
  booktitle={Journal of nonlinear mathematical physics},
  volume={24},
  number={4},
  pages={584-595},
  year={2017},
}
Xiang-Ke Chang, and Jacek Szmigielski. Liouville integrability of conservative peakons for a modified CH equation. 2017. Vol. 24. In Journal of nonlinear mathematical physics. pp.584-595. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221112726491645732.
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