Journal of Evolution Equations

, Volume 16, Issue 3, pp 539–567 | Cite as

A new highly nonlinear shallow water wave equation

Article

Abstract

We derive a quasilinear shallow water equation directly from the governing equations for gravity water waves within a certain regime for large-amplitude waves which has not been studied so far. Furthermore, we demonstrate local well-posedness of the corresponding Cauchy problem and finally discuss some aspects of the blowup behavior of solutions.

Keywords

Shallow water equation Large-amplitude waves Cubic nonlinearity Well-posedness 

Mathematics Subject Classification

Primary 35Q35 Secondary 35L30 

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References

  1. 1.
    Benjamin T. B., Bona J. L., Mahoney J. J.: Model equations for long waves in nonlinear dispersive systems. Phil. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 227, 47–78 (1972)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Camassa R., Holm D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    A. Constantin Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis CBMS-NSF Regional Conference Series in Applied Mathematics 81, SIAM, Philadelphia (2011).Google Scholar
  4. 4.
    Constantin A., Escher J.: Global Existence and Blow-up for a Shallow Water Equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci 26(4), 303–328 (1998)MathSciNetMATHGoogle Scholar
  5. 5.
    Constantin A., Escher J.: Well-Posedness, Global Existence, and Blowup Phenomena for a Periodic Quasi-Linear Hyperbolic Equation. Comm. Pure Appl. Math. 61, 475–504 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Constantin A., Escher J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Constantin A., Ivanov R. I., Lenells J.: Inverse scattering transform for the Degasperis–Procesi equation. Nonlinearity 23, 2559–2575 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Constantin A., Johnson R. S.: On the Non-Dimensionalisation, Scaling and Resulting Interpretation of the Classical Governing Equations for Water Waves. J. Nonlinear Math. Phys. 15, 58–73 (2008)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Constantin A., Lannes D.: The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations. Arch. Rational Mech. Anal. 192, 165–186 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Constantin A., McKean H. P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Degasperis A., Holm D. D., Hone A. N. W.: A new integrable equation with peakon solutions. Theor. Math. Phys. 133, 1463–1474 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Drazin P. G., Johnson R. S.: Solitons: an introduction. Cambridge Univ. Press, Cambridge (1990)MATHGoogle Scholar
  13. 13.
    Degasperis A., Procesi M.: Asymptotic integrability In: Degasperis, A. and Gaeta, G. (eds.) Symmetry and Perturbation Theory, pp. 23–37. World Scientific, Singapore (1999).Google Scholar
  14. 14.
    Escher J., Liu Y., Yin D.: Global weak solutions and blow-up structure for the Degasperis-Procesi equation. J. Funct. Analysis 192, 457–485 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Fuchssteiner B., Fokas A. S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D 4, 47–66 (1981)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ionescu-Kruse D.: Variational derivation of the Camassa–Holm shallow water equation. J. Nonlinear Math. Phys. 14, 303–312 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Ivanov R. I.: On the Integrability of a Class of Nonlinear Dispersive Wave Equations. J. Nonlinear Math. Phys. 12, 462–468 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ivanov R. I.: Water waves and integrability. Philos. Trans. Roy. Soc. London A 365, 2267–2280 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Johnson R. S.: Camassa–Holm, Korteweg–de Vries and related models for water waves J. Fluid Mech. 455, 63–82 (2002)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Johnson R. S.: The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations. J. Nonlinear Math. Phys. 10, 72–92 (2003)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Johnson R. S.: A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge Univ. Press, Cambridge (1997)CrossRefMATHGoogle Scholar
  22. 22.
    Kato T.: On the Korteweg–De Vries Equation. Manuscripta Math. 28, 89–100 (1979)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    T. Kato Quasi-linear equations of evolution, with applications to partial differential equations In: Spectral Theory and Differential Equations, pp. 25-70. Springer Lecture Notes in Mathematics 448, Berlin (1975).Google Scholar
  24. 24.
    Lannes D.: The Water Waves Problem: Mathematical Analysis and Asymptotics, American Math. Soc., Providence, RI (2013)CrossRefMATHGoogle Scholar
  25. 25.
    Mutlubaş N. D.: Local well-posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude. Nonlinear Anal. R. World Appl. 97, 145–154 (2014)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefMATHGoogle Scholar
  27. 27.
    Stein E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton (1993)MATHGoogle Scholar
  28. 28.
    T. Tao, Low-regularity global solutions to nonlinear dispersive equations In: Surveys in Analysis and Operator Theory, pp. 19–48, Proc. Centre Math. Appl. Austral. Nat. Univ. 40 (2002).Google Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria

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