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Wave Breaking for a Nonlinear Shallow Water Equation

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Abstract

In this paper, we mainly study wave breaking for a nonlinear shallow water equation which describes the surface water waves of moderate amplitude in shallow water regime. We present a sufficient condition on the initial data to lead to wave breaking. Moreover, the estimate of life span is derived.

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References

  1. Camassa, R., Holm, D.D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71(11), 1661–1664 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  2. Constantin, A.: Existence of permanent and breaking waves for a shallow water equation: a geometric approach. Annales- Institut Fourier 50, 321–362 (2000)

    Article  MathSciNet  Google Scholar 

  3. Constantin, A.: Nonlinear water waves with applications to wave-current interactions and tsunamis. International Journal of Geometric Methods in Modern Physics 6(6), 419–450 (2011)

    Google Scholar 

  4. Constantin, A., Escher, J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181(2), 229–243 (1998)

    Article  MathSciNet  Google Scholar 

  5. Constantin, A., Escher, J.: Global existence and blow-up for a shallow water equation. Annali Della Scuola Normale Superiore Di Pis 26(2), 303–328 (1998)

    MathSciNet  MATH  Google Scholar 

  6. Constantin, A., Escher, J.: On the blow-up rate and the blow-up set of breaking waves for a shallow water equation. Math. Z. 233(1), 75–91 (2000)

    Article  MathSciNet  Google Scholar 

  7. Constantin, A., Lannes, D.: The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 192(1), 165–186 (2007)

    Article  MathSciNet  Google Scholar 

  8. Fisher, M., Schiff, J.: The Camassa-Holm equation: conserved quantities and the initial value problem. Phys. Lett. A 259(5), 371–376 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  9. Fuchssteiner, B., Fokas, A.S.: Symplectic structures, their Bcklund transformations and hereditary symmetries. Physica D 4(1), 47–66 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  10. Gasull, A., Geyer, A.: Traveling surface waves of moderate amplitude in shallow water. Nonlinear Anal. Theory Methods Appl. 102(100), 105–119 (2014)

    Article  MathSciNet  Google Scholar 

  11. Geyer, A.: Solitary traveling water waves of moderate amplitude. Journal of Nonlinear Mathematical Physics 19(sup1), 1–12 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Geyer, A.: Symmetric waves are traveling waves for a shallow water equation modeling surface waves of moderate amplitude. Journal of Nonlinear Mathematical Physics 22(4), 545–551 (2015)

    Article  MathSciNet  Google Scholar 

  13. Johnson, R.S.: Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455, 63–82 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  14. Ma, F., Yue, L., Qu, C.: Wave-breaking phenomena for the nonlocal Whitham-type equations. J. Differential Equations 261(11), 6029–6054 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  15. Mi, Y., Mu, C.: On the solutions of a model equation for shallow water waves of moderate amplitude. J. Differential Equations 8(8), 2101–2129 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  16. Mutluba, D.: On the Cauchy problem for a model equation for shallow water waves of moderate amplitude. Nonlinear Anal. Real World Appl. 14(5), 2022–2026 (2013)

    Article  MathSciNet  Google Scholar 

  17. Mutluba, D.: Local well-posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude. Nonlinear Anal. Theory Methods Appl. 97, 145–154 (2014)

    Article  MathSciNet  Google Scholar 

  18. Mutluba, D., Geyer, A.: Orbital stability of solitary waves of moderate amplitude in shallow water. J. Differential Equations 255(2), 254–263 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  19. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (2011)

    MATH  Google Scholar 

  20. Yang, S.: Wave breaking for a model equation for shallow water waves of moderate amplitude. Commun. Math. Sci. 19(7), 1799–1807 (2021)

    Article  MathSciNet  Google Scholar 

  21. Zhou, S.: The local well-posedness, existence and uniqueness of weak solutions for a model equation for shallow water waves of moderate amplitude. J. Differential Equations 258(12), 4103–4126 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  22. Zhou, S., Mu, C.: Global conservative solutions for a model equation for shallow water waves of moderate amplitude. J. Differential Equations 256(5), 1793–1816 (2014)

    Article  MathSciNet  Google Scholar 

  23. Zhou, S., Qiao, Z., Mu, C., et al.: Continuity and asymptotic behaviors for a shallow water wave model with moderate amplitude. J. Differential Equations 263(2), 910–933 (2017)

    Article  ADS  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by Yunnan Fundamental Research Projects (Grant NO. 202101AU070029).

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Authors and Affiliations

  1. Department of Systems Science and Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan, 650500, China

    Jianmin Zhao & Shaojie Yang

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  1. Jianmin Zhao
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  2. Shaojie Yang
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Correspondence to Shaojie Yang.

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Communicated by A. Constantin.

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Zhao, J., Yang, S. Wave Breaking for a Nonlinear Shallow Water Equation. J. Math. Fluid Mech. 24, 89 (2022). https://doi.org/10.1007/s00021-022-00724-0

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  • DOI: https://doi.org/10.1007/s00021-022-00724-0

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