Advertisement

Journal of Evolution Equations

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

A new highly nonlinear shallow water wave equation

  • Ronald Quirchmayr
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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Camassa R., Holm D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Constantin A., Escher J.: Wave breaking for nonlinear nonlocal shallow water equations. Acta Math. 181, 229–243 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Constantin A., Ivanov R. I., Lenells J.: Inverse scattering transform for the Degasperis–Procesi equation. Nonlinearity 23, 2559–2575 (2010)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Constantin A., McKean H. P.: A shallow water equation on the circle. Comm. Pure Appl. Math. 52, 949–982 (1999)MathSciNetCrossRefzbMATHGoogle 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)zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fuchssteiner B., Fokas A. S.: Symplectic structures, their Bäcklund transformations and hereditary symmetries. Physica D 4, 47–66 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ionescu-Kruse D.: Variational derivation of the Camassa–Holm shallow water equation. J. Nonlinear Math. Phys. 14, 303–312 (2007)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ivanov R. I.: Water waves and integrability. Philos. Trans. Roy. Soc. London A 365, 2267–2280 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Johnson R. S.: Camassa–Holm, Korteweg–de Vries and related models for water waves J. Fluid Mech. 455, 63–82 (2002)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  22. 22.
    Kato T.: On the Korteweg–De Vries Equation. Manuscripta Math. 28, 89–100 (1979)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  27. 27.
    Stein E.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Univ. Press, Princeton (1993)zbMATHGoogle 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

Personalised recommendations