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Well-posedness and continuity properties of the new shallow-water model with cubic nonlinearity

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Abstract

This paper is devoted to the new shallow-water model (also called the modified Camassa–Holm–Novikov equation) with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the Fokas–Olver–Rosenau–Qiao equation and the Novikov equation as two special cases. It is shown that the Cauchy problem of the modified Camassa–Holm–Novikov equation for the periodic and the nonperiodic case is well-posed in Sobolev spaces in the sense of Hadamard, that is, the data-to-solution map is continuous. However, the solution map is not uniformly continuous.

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Acknowledgements

The work of Mi is partially supported by NSF of China (11671055). The work of Huang is partially supported by NSF of China (11971067, 11631008, 11771183).

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Authors and Affiliations

  1. College of Mathematics and Computer Sciences, Yangtze Normal University, Fuling, Chongqing, 408100, People’s Republic of China

    Yongsheng Mi

  2. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, People’s Republic of China

    Daiwen Huang

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  1. Yongsheng Mi
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Correspondence to Yongsheng Mi.

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Mi, Y., Huang, D. Well-posedness and continuity properties of the new shallow-water model with cubic nonlinearity. Annali di Matematica 200, 1–34 (2021). https://doi.org/10.1007/s10231-020-00980-9

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  • DOI: https://doi.org/10.1007/s10231-020-00980-9

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