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Stability of Solitary Waves for the Modified Camassa-Holm Equation

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Abstract

We study the stability of smooth and peaked solitary waves to the modified Camassa-Holm equation. This quasilinear equation with cubic nonlinearity is completely integrable and arises as a model for the unidirectional propagation of shallow water waves. Based on the phase portrait analysis, we demonstrate the existence of unique localized smooth solcontra1itary-wave solution with certain range of the linear dispersive parameter. We then show orbital stability of the smooth solitary-wave solution under small disturbances by means of variational methods, considering a minimization problem with an appropriate constraint. Using the variational approach with suitable conservation laws, we also establish the orbital stability of peakons in the Sobolev space \( H^1 \cap W^{1, 4} \) without the assumption on the positive momentum density initially. Finally we demonstrate spectral stability of such smooth solitary waves using refined spectral analysis of the linear operator corresponding to the second-order variational derivative of the local Hamiltonian.

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Acknowledgements

The work of Li is partially supported by the NSFC grant 11771161. The work of Liu is partially supported by the Simons Foundation grant 499875.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Ji Li

  2. Department of Mathematics, University of Texas at Arlington, Arlington, TX, 76019, USA

    Yue Liu

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  1. Ji Li
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  2. Yue Liu
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Correspondence to Yue Liu.

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Li, J., Liu, Y. Stability of Solitary Waves for the Modified Camassa-Holm Equation. Ann. PDE 7, 14 (2021). https://doi.org/10.1007/s40818-021-00104-3

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