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arXiv · 2412.04124

On a two-component Camassa-Holm equation

Zixing Zhang·Q. P. Liu
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Abstract

A two-component generalization of the Camassa-Holm equation and its reduction proposed recently by Xue, Du and Geng [Appl. Math. Lett. {\bf 146} (2023) 108795] are studied. For this two-component equation, its missing bi-Hamiltonian structure is constructed and a Miura transformation is introduced so that it may be regarded as a modification of the very first two-component Camassa-Holm equation. %[Phys. Rev. E {\bf 53} (1996) ; Lett. Math. Phys. {\bf 53 } (2006)]. Using a proper reciprocal transformation, a particular reduction of this two-component equation, which admits $N-$ peakon solution, is brought to the celebrated Burgers equation.

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Zixing Zhang, Q. P. Liu. 2024-12-05. On a two-component Camassa-Holm equation. https://arxiv.org/abs/2412.04124

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