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Blow-up and peakons for a higher-order \(\mu \)-Camassa–Holm equation

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Abstract

This paper proposes a higher-order \(\mu \)-Camassa–Holm equation, which is regarded as a higher-order extension of the \(\mu \)-Camassa–Holm equation, and preserves some properties of the \(\mu \)-Camassa–Holm equation. We first show that the equation admits the peaked traveling wave solution, which is given by a Green function of the momentum operator. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established. Finally, the blow-up criterion and wave breaking mechanism for solutions with certain initial profiles are studied. It turns out that all the nonlinearities even the first-order nonlinearity may have the effect on the blow up.

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Acknowledgements

The work of Luo is supported by the NSF of Zhejiang Province Grant LQ20A010006 and NSF-China Grant-12001491. The work of Fu is supported by the National Science Basic Research Program of Shaanxi Grant-2020JC-37. The work of Qu is supported by the NSF of China Grant-11631007 and Grant-11971251.

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Authors and Affiliations

  1. School of Science, Najing University of Science and Technology, Nanjing, 210014, People’s Republic of China

    Hao Wang

  2. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Ting Luo

  3. School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi’an, 710127, People’s Republic of China

    Ying Fu

  4. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Changzheng Qu

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  1. Hao Wang
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  4. Changzheng Qu
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Correspondence to Ting Luo.

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Wang, H., Luo, T., Fu, Y. et al. Blow-up and peakons for a higher-order \(\mu \)-Camassa–Holm equation. J. Evol. Equ. 22, 13 (2022). https://doi.org/10.1007/s00028-022-00774-x

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