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Wave-breaking analysis and weak multi-peakon solutions for a generalized cubic–quintic Camassa–Holm type equation

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Abstract

We consider the Cauchy problem and multi-peakon solutions of a generalized cubic–quintic Camassa–Holm (gcqCH) equation, which is actually an extension of the cubic CH equation [alias the Fokas–Olver–Rosenau–Qiao equation in the literature] and the quintic CH equation, and possesses the Hamiltonian structure and conversation law. We first present the blow-up criteria and the precise blow-up quantity in terms of the Moser-type estimate in Sobolev spaces. Then, by using the blow-up quantity and the characteristics associated with the gcqCH equation, we obtain two kinds of sufficient conditions on the initial data to guarantee the occurrence of the wave-breaking phenomenon. Finally, the non-periodic and periodic peakon as well as global N-peakon solutions of the gcqCH equation are also investigated. Particularly, we study the two-peakon dynamical system with the time evolution of their elastic collisions.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11925108 and 11971475). The second author also thanks the UT President’s Endowed Professorship (Project # 450000123) for its partial support.

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Authors and Affiliations

  1. Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Weifang Weng & Zhenya Yan

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Weifang Weng & Zhenya Yan

  3. School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX, USA

    Zhijun Qiao

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  1. Weifang Weng
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Correspondence to Zhenya Yan.

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Communicated by Adrian Constantin.

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Weng, W., Qiao, Z. & Yan, Z. Wave-breaking analysis and weak multi-peakon solutions for a generalized cubic–quintic Camassa–Holm type equation. Monatsh Math 200, 667–713 (2023). https://doi.org/10.1007/s00605-022-01699-w

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