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International Conference on Intelligent Computing

ICIC 2007: Advanced Intelligent Computing Theories and Applications. With Aspects of Artificial Intelligence pp 1287–1295Cite as

A Difference Scheme for the Camassa-Holm Equation

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 4682))

Abstract

In this paper, a difference scheme satisfying two conservative laws for the periodic Camassa-Holm equation is given. Convergence is proved. The numerical simulation exhibits the time evolution and the interaction of solitary wave solutions.

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Constantin, A., Escher, J.: Global weak solutions for a shallow water equation. Indiana Univ. Math. J. 47, 1527–1545 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Constantin, A., Escher, J.: Well-posedness, global existence, and blow-up phenomena for a periodic quasi-linear hyperbolic equation. Comm. Pure Appl. Math. 51, 475–504 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Constantin, A.: On the blow-up of solutions a periodic shallow water equation. J. Nonlinear Sci. 10, 391–399 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Comm. Math. Phys. 211, 45–61 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Constantin, A., Strauss, W.A.: Stability of peakons. Commun. Pure Appl. Math. LIII, 603–610 (2000)

    Article  MathSciNet  Google Scholar 

  7. Constantin, A., Molinet, L.: Global weak solutions for a shallow water equation. Comm. Math. Phys. 211, 45–61 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Constantin, A., Strauss, W.A.: Stability of the Camassa-Holm solitons. J. Nonlinear. Sci. 12, 415–422 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. McKean, H.P.: Breakdown of the Camassa-Holm equation. Comm. Pure Appl. Math. LVII, 416–418 (2004)

    Article  MathSciNet  Google Scholar 

  10. Kalisch, H., Lenells, J.: Numerical study of traveling-wave solutions for the Camassa-Hlom equation. Solitons and Fractals 25, 287–298 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Artebrant, R., Schroll, H.J.: Numerical simulation of Camassa-Holm peakons by adaptive upwinding. Appl. Numer. Math. 56, 695–711 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Zhang, L.M., Chang, Q.S.: A conservative numerical scheme for a class of nonliner Schrodinger equation with wave operator. Appl. Math. Comput. 145, 603–612 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ablowitz, M.J., Clarkson, P.A.: Solitons, nonlinear evolution equations and inverse scattering. London Mathematical Society, Lecture Note Series, vol. 149. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, P.R. China

    Ahamed Adam Abdelgadir, Yang-xin Yao, Yi-ping Fu & Ping Huang

Authors
  1. Ahamed Adam Abdelgadir
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  2. Yang-xin Yao
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  3. Yi-ping Fu
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  4. Ping Huang
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Editor information

De-Shuang Huang Laurent Heutte Marco Loog

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© 2007 Springer-Verlag Berlin Heidelberg

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Abdelgadir, A.A., Yao, Yx., Fu, Yp., Huang, P. (2007). A Difference Scheme for the Camassa-Holm Equation. In: Huang, DS., Heutte, L., Loog, M. (eds) Advanced Intelligent Computing Theories and Applications. With Aspects of Artificial Intelligence. ICIC 2007. Lecture Notes in Computer Science(), vol 4682. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74205-0_132

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  • DOI: https://doi.org/10.1007/978-3-540-74205-0_132

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74201-2

  • Online ISBN: 978-3-540-74205-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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