Skip to main content
Log in

Peaked wave solutions of CH-r equation

  • Science in China (Series A)
  • Published:
Science in China Series A: Mathematics Aims and scope Submit manuscript

Abstract

In this paper the qualitative analysis methods of planar autonomous systems and numerical simulation are used to investigate the peaked wave solutions of CH-r equation. Some explicit expressions of peaked solitary wave solutions and peaked periodic wave solutions are obtained, and some of their relationships are revealed. Why peaked points are generated is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Boyd, J. P., Peakons and coshoidal waves: travelling wave solutions of the Camassa-Holm equation, Appl. Math. Comput., 1997, 81(2–3): 173–187.

    Article  MATH  MathSciNet  Google Scholar 

  3. Cooper, F., Shepard, H., Solitons in the Camassa-Holm shallow water equation, Phys. Lett. A, 1994, 194(4): 246–250.

    Article  MATH  MathSciNet  Google Scholar 

  4. Constantin, A., Soliton interactions for the Camassa-Holm equation, Exposition Math., 1997, 15(3): 251–264.

    MATH  MathSciNet  Google Scholar 

  5. Liu Zhengrong, Qian Tifei, Peakons of the Camassa-Holm equation, Applied Mathematical Modelling, 2002, 26: 473–480.

    Article  MATH  Google Scholar 

  6. Dullin, H. R., Gottwald, G. A., Holm, D. D., An integrable shallow water equation with linear and nonlinear dispersion, to appear.

  7. Holm, D. D., Notes on CH-r equation: Local and nonlocal, dimensional and nondimensional, to appear.

  8. Constantin, A., Quasi-periodicity with respect to time of spatially periodic finite-gap solution of the Camassa-Holm equation, Bull. Sci. Math., 1998, 127(7): 487–494.

    Article  MathSciNet  Google Scholar 

  9. Constantin, A., On the Cauchy problem for the periodic Camassa-Holm equation, J. Differential Equations, 1997, 141(2): 218–235.

    Article  MATH  MathSciNet  Google Scholar 

  10. Dai, H. H., Pavlov, M., Maxim transformations for the Camassa-Holm equation, its high-frequency limit and the Sinh-Gordon equation, J. Phys. Soc. Jpn., 1998, 67(11): 3655–3657.

    Article  MATH  MathSciNet  Google Scholar 

  11. Fuchssteiner, B., Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm equation, Phys. D, 1996, 95(3–4): 229–243.

    Article  MATH  MathSciNet  Google Scholar 

  12. Kouranbaeva, S., The Camassa-Holm equation as a geodesic flow on the diffeomorphism group, J. Math. Phys., 1999, 40(2): 857–868.

    Article  MATH  MathSciNet  Google Scholar 

  13. Schiff, J., The Camassa-Holm equation: a loop group approach, Phys. D, 1998, 121(1–2): 24–43.

    Article  MATH  MathSciNet  Google Scholar 

  14. Qian Tifei, Tang Minying, Peakons and periodic cusp waves in a generalized Camassa-Holm equation, Choas, Solitons and Fractals, 2001, 12: 1347–1360.

    Article  MATH  Google Scholar 

  15. Liu Zhengrong, Qian Tifei, Peakons and their bifurcation in a generalized Camassa-Holm equation, International Journal of Bifurcation and Choas, 2001, 11(3): 781–792.

    Article  MATH  Google Scholar 

  16. Alber, M. S., Camassa, R., Fedorov, Y. N. et al., The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE’s of shallow water and Dym type, Comm. Math. Phys., 2001, 221(1): 197–227.

    Article  MATH  MathSciNet  Google Scholar 

  17. Reyes, E. G., Geometric integrability of thd Camassa-Holm equation, Lett. Math. Phys., 2002, 59(2): 117–131.

    Article  MATH  MathSciNet  Google Scholar 

  18. Foias, C., Holm, D. D., Titi, E. S., The three dimenional viscous Camassa-Holm equation, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations, 2002, 14(1): 1–35.

    Article  MATH  MathSciNet  Google Scholar 

  19. Korteweg, D. J., de Vriss, G., On the change of form long wave advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. Ser., 1895, 39(5): 422–443.

    Google Scholar 

  20. Guckenheimer, J., Holmes, P., Dynamical Systems and Bifurcation of Vector Fields, New York: Springer-Verlag, 1983.

    Google Scholar 

  21. Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos, New York: Springer-Verlag, 1990.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Boling Guo.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guo, B., Liu, Z. Peaked wave solutions of CH-r equation. Sci. China Ser. A-Math. 46, 696–709 (2003). https://doi.org/10.1007/BF02942241

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02942241

Keywords

Navigation