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Long Time Behaviour for a Class of Low-Regularity Solutions of the Camassa-Holm Equation

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In this paper, we investigate the long time behaviour for a class of low- regularity solutions of the Camasssa-Holm equation given by the superposition of infinitely many interacting traveling waves with corners at their peaks.

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References

  1. Beals R., Sattinger D., Szmigielski J.: Multipeakons and the classical moment problem. Adv Math. 154, 229–257 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bressan A., Constantin A.: Global conservative solutions of the Camassa-Holm equation. Arch. Rat. Mech. Anal. 183, 215–239 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bressan A., Constantin A.: Global dissipative solutions of the Camassa-Holm equation. Anal. Appl. (Singap.) 5, 1–27 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  5. Camassa R., Holm D., Hyman J.M.: A new integrable shallow water equation. Adv. Appl. Mech. 31, 1–33 (1994)

    Article  Google Scholar 

  6. Camassa R., Huang J., Lee L.: On a completely integrable numerical scheme for a nonlinear shallow-water equation. J. Nonlinear Math. Phys. 12(Supplement 1), 146–152 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  7. Camassa R., Huang J., Lee L.: Integral and integrable algorithms for a nonlinear shallow-water equation. J. Comput. Phys. 216, 547–572 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Calogero F., Francoise J.-P.: A completely integrable Hamiltonian system. J. Math. Phys. 37, 2863–2871 (1996)

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. Constantin A., Strauss W.: Stability of peakons. Comm. Pure Appl. Math. 53(5), 603–610 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. De Lellis C., Kappeler T., Topalov P.: Low-regularity solutions of the periodic Camassa-Holm equation. Comm. Part. Differ. Eq. 32, 87–126 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Deift P., Li L.-C., Tomei C.: Toda flows with infinitely many variables. J. Funct. Anal. 64, 358–402 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  13. Deift P., Li L.-C., Tomei C.: Matrix factorizations and integrable systems. Comm. Pure Appl. Math. 42, 443–521 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  14. El Dika, K., Molinet, L.: Stability of multipeakons. http://arXiv.org/list/0803.0261.v1[math.AP], 2008

  15. Gantmacher, F., Krein, M.: Oscillation matrices and kernels and small vibrations of mechanical systems. Revised edition. AMS Chelsea Publishing, Providence, R.I.:Amer. Math. Soc., 2002

  16. Holden H., Raynaud X.: A convergent numerical scheme for the Camassa-Holm equation based on multipeakons. Discrete Contin. Dyn. Syst. 14, 505–523 (2006)

    MATH  MathSciNet  Google Scholar 

  17. Li L.-C.: Long time behaviour of an infinite particle system. Commun. Math. Phys. 110, 617–623 (1987)

    Article  MATH  ADS  Google Scholar 

  18. Li L.-C.: Factorization problem on the Hilbert-Schmidt group and the Camassa-Holm equation. Comm. Pure Appl. Math. 61, 186–209 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Loomis L., Sternberg S.: Advanced calculus. Addision Wesley, Reading, MA (1968)

    MATH  Google Scholar 

  20. McKean H.: Fredholm determinants and the Camassa-Holm hierarchy. Comm. Pure Appl. Math. 56, 638–680 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Moser, J.: Finitely many mass points on the line under the influence of an exponential potential-an integrable system. In: Dynamical systems, theory and applications, Lecture Notes in Physics, Vol. 38, Berlin: Springer-Verlag, 1975, pp. 467–497

  22. Ragnisco O., Bruschi M.: Peakons, r-matrix and Toda lattice. Phys. A 228, 150–159 (1996)

    Article  MathSciNet  Google Scholar 

  23. Reed M., Simon B.: Methods of mathematical physics I: Functional analysis. Academic Press, New York- London (1972)

    MATH  Google Scholar 

  24. Reed M., Simon B.: Methods of mathematical physics IV: Analysis of operators. Academic Press, New York-London (1972)

    Google Scholar 

  25. Reyman, A., Semenov-Tian-Shansky, M.: Group-theoretical methods in the theory of finite- dimensional integrable systems. Dynamical Systems VII, Encyclopaedia of Mathematical Sciences, V.I. Arnold, S.P. Novikov, eds., Vol. 16, Berlin-Heidelberg-NewYork: Springer- Verlag, 1994, 116–225

  26. Semenov-Tian-Shansky, M.: Classical r-matrices, Lax equations, Poisson Lie groups and dressing transformations. In: Field theory, quantum gravity and strings II (Meudon/Paris, 1985/1986) Lecture Notes in Physics, Vol 280, Berlin-Heidelberg-NewYork: Springer-Verlag 1987, pp. 174–214

  27. Smithies F.: Integral equations Cambridge tracts in Mathematics and Mathematical Physics, no. 49. Cambridge University Press, Cambridge (1958)

    Google Scholar 

  28. Stieltjes T.J.: Sur la réduction en fraction continue d’une série procédant suivant les puissances descendantes d’une variable. Ann. Fac. Sci. Toulouse Math. 3, H1–H17 (1889)

    MathSciNet  Google Scholar 

  29. Xin Z., Zhang P.: On the weak solutions to a shallow water equation. Comm. Pure Appl. Math. 53, 1411–1433 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics, Pennsylvania State University, University Park, PA, 16802, USA

    Luen-Chau Li

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  1. Luen-Chau Li
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Correspondence to Luen-Chau Li.

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Communicated by P. Constantin

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Li, LC. Long Time Behaviour for a Class of Low-Regularity Solutions of the Camassa-Holm Equation. Commun. Math. Phys. 285, 265–291 (2009). https://doi.org/10.1007/s00220-008-0603-5

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