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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 164876, 17 pages
http://dx.doi.org/10.1155/2012/164876
Research Article

On the Study of Local Solutions for a Generalized Camassa-Holm Equation

School of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China

Received 23 May 2012; Revised 30 June 2012; Accepted 18 July 2012

Academic Editor: Yong Hong Wu

Copyright © 2012 Meng Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993. View at Publisher · View at Google Scholar
  2. S. Y. Lai and Y. H. Wu, “A model containing both the Camassa-Holm and Degasperis-Procesi equations,” Journal of Mathematical Analysis and Applications, vol. 374, no. 2, pp. 458–469, 2011. View at Publisher · View at Google Scholar
  3. S. Y. Lai, Q. C. Xie, Y. X. Guo, and Y. H. Wu, “The existence of weak solutions for a generalized Camassa-Holm equation,” Communications on Pure and Applied Analysis, vol. 10, no. 1, pp. 45–57, 2011. View at Publisher · View at Google Scholar
  4. S. Y. Lai and Y. H. Wu, “Existence of weak solutions in lower order Sobolev space for a Camassa-Holm-type equation,” Journal of Physics A, vol. 43, no. 9, Article ID 095205, 13 pages, 2010. View at Publisher · View at Google Scholar
  5. X. G. Li, Y. H. Wu, and S. Y. Lai, “A sharp threshold of blow-up for coupled nonlinear Schrödinger equations,” Journal of Physics A, vol. 43, no. 16, Article ID 165205, 11 pages, 2010. View at Publisher · View at Google Scholar
  6. J. Zhang, X. G. Li, and Y. H. Wu, “Remarks on the blow-up rate for critical nonlinear Schrödinger equation with harmonic potential,” Applied Mathematics and Computation, vol. 208, no. 2, pp. 389–396, 2009. View at Publisher · View at Google Scholar
  7. A. Constantin and H. P. McKean, “A shallow water equation on the circle,” Communications on Pure and Applied Mathematics, vol. 52, no. 8, pp. 949–982, 1999. View at Publisher · View at Google Scholar
  8. A. Constantin, “On the inverse spectral problem for the Camassa-Holm equation,” Journal of Functional Analysis, vol. 155, no. 2, pp. 352–363, 1998. View at Publisher · View at Google Scholar
  9. A. Constantin, V. S. Gerdjikov, and R. I. Ivanov, “Inverse scattering transform for the Camassa-Holm equation,” Inverse Problems, vol. 22, no. 6, pp. 2197–2207, 2006. View at Publisher · View at Google Scholar
  10. H. P. McKean, “Integrable systems and algebraic curves,” in Global Analysis (Proceedings of the Biennial Seminar of the Canadian Mathematical Congress, University of Calgary, Calgary, Alberta, 1978), vol. 755 of Lecture Notes in Mathematics, pp. 83–200, Springer, Berlin, Germany, 1979.
  11. A. Constantin, T. Kappeler, B. Kolev, and P. Topalov, “On geodesic exponential maps of the Virasoro group,” Annals of Global Analysis and Geometry, vol. 31, no. 2, pp. 155–180, 2007. View at Publisher · View at Google Scholar
  12. G. A. Misiołek, “A shallow water equation as a geodesic flow on the Bott-Virasoro group,” Journal of Geometry and Physics, vol. 24, no. 3, pp. 203–208, 1998. View at Publisher · View at Google Scholar
  13. Y. A. Li and P. J. Olver, “Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,” Journal of Differential Equations, vol. 162, no. 1, pp. 27–63, 2000. View at Publisher · View at Google Scholar
  14. A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229–243, 1998. View at Publisher · View at Google Scholar
  15. S. Hakkaev and K. Kirchev, “Local well-posedness and orbital stability of solitary wave solutions for the generalized Camassa-Holm equation,” Communications in Partial Differential Equations, vol. 30, no. 4–6, pp. 761–781, 2005. View at Publisher · View at Google Scholar
  16. S. Y. Lai and Y. H. Wu, “The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,” Journal of Differential Equations, vol. 248, no. 8, pp. 2038–2063, 2010. View at Publisher · View at Google Scholar
  17. N. Li, S. Y. Lai, S. Li, and M. Wu, “The local and global existence of solutions for a generalized Camassa-Holm equation,” Abstract and Applied Analysis, vol. 2012, Article ID 532369, 27 pages, 2012. View at Publisher · View at Google Scholar
  18. J. L. Bona and R. Smith, “The initial-value problem for the Korteweg-de Vries equation,” Philosophical Transactions of the Royal Society of London, Series A, vol. 278, no. 1287, pp. 555–601, 1975.
  19. G. Rodríguez-Blanco, “On the Cauchy problem for the Camassa-Holm equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 46, no. 3, pp. 309–327, 2001. View at Publisher · View at Google Scholar
  20. T. Kato, “Quasi-linear equations of evolution, with applications to partial differential equations,” in Spectral Theory and Differential Equations, Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.