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On the Cauchy Problem for the New Shallow-water Models with Cubic Nonlinearity

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Abstract

This paper is devoted to the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa–Holm equation (also called Fokas–Olver–Rosenau–Qiao equation) and the Novikov equation as two special cases. On the one hand, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions in the Gevrey–Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map for the system. On the other hand, we prove the persistence properties in weighted spaces of the solution, provided that the initial potential satisfies a certain sign condition.

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Acknowledgements

The work was supported by NSF of Chongqing (Nos. cstc2020jcyj-jqX0025, CSTB2023NSCQ-LZX0035).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Yangtze Normal University, Chongqing, 408100, China

    Yongsheng Mi

  2. Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Boling Guo

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  1. Yongsheng Mi
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Correspondence to Yongsheng Mi.

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Mi, Y., Guo, B. On the Cauchy Problem for the New Shallow-water Models with Cubic Nonlinearity. Front. Math 19, 435–455 (2024). https://doi.org/10.1007/s11464-021-0319-9

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  • DOI: https://doi.org/10.1007/s11464-021-0319-9

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