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Stability of the Camassa-Holm peakons in the dynamics of a shallow-water-type system

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Abstract

The stability of the Camassa-Holm (periodic) peakons in the dynamics of an integrable shallow-water-type system is investigated. A variational approach with the use of the Lyapunov method is presented to prove the variational characterization and the orbital stability of these wave patterns. In addition, a sufficient condition for the global existence of strong solutions is given. Finally, a local-in-space wave-breaking criterion is illustrated in the periodic setting.

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Acknowledgments

The work of R.M. Chen was partially supported by the Central Research Development Fund No. 04.13205.30205 from University of Pittsburgh. The work of X.C. Liu is supported by NSF-China grant 11401471 and Ph.D. Programs Foundation of Ministry of Education of China-20136101120017. The work of Y. Liu is partially supported by NSF grant DMS-1207840. The work of C.Z. Qu is supported by the NSF-China grant-11471174 and NSF of Ningbo grant-2014A610018.

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Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Robin Ming Chen

  2. Center for Nonlinear Studies and School of Mathematics, Northwest University, Xi’an, 710069, China

    Xiaochuan Liu

  3. Department of Mathematics, University of Texas at Arlington, Arlington, TX, 76019-0408, USA

    Yue Liu

  4. Department of Mathematics, Ningbo University, Ningbo, 315211, China

    Robin Ming Chen, Yue Liu & Changzheng Qu

Authors
  1. Robin Ming Chen
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  2. Xiaochuan Liu
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  3. Yue Liu
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  4. Changzheng Qu
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Correspondence to Robin Ming Chen.

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Communicated by P. Rabinowitz.

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Chen, R.M., Liu, X., Liu, Y. et al. Stability of the Camassa-Holm peakons in the dynamics of a shallow-water-type system. Calc. Var. 55, 34 (2016). https://doi.org/10.1007/s00526-016-0972-0

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