Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2014, Article ID 348695, 16 pages
http://dx.doi.org/10.1155/2014/348695
Research Article

On the Global Dissipative and Multipeakon Dissipative Behavior of the Two-Component Camassa-Holm System

1School of Automation, Chongqing University, Chongqing 400044, China
2Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway

Received 14 February 2014; Revised 20 March 2014; Accepted 20 March 2014; Published 4 May 2014

Academic Editor: Bo Shen

Copyright © 2014 Yujuan Wang et al. 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. 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 · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. 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 · View at Zentralblatt MATH · View at Scopus
  3. A. Constantin, “The Hamiltonian structure of the Camassa-Holm equation,” Expositiones Mathematicae, vol. 15, no. 1, pp. 53–85, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. A. Constantin, “On the scattering problem for the Camassa-Holm equation,” Proceedings of the the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, vol. 457, no. 2008, pp. 953–970, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. Constantin, “Global existence of solutions and breaking waves for a shallow water equation: a geometric approach,” Annales de l'Institut Fourier, vol. 50, no. 2, pp. 321–362, 2000. View at Publisher · View at Google Scholar
  6. A. Constantin and D. Lannes, “The hydrodynamical relevance of the Camassa–Holm and Degasperis–Procesi equations,” Archive for Rational Mechanics and Analysis, vol. 192, no. 1, pp. 165–186, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. A. Bressan and A. Constantin, “Global conservative solutions of the Camassa–Holm equation,” Archive for Rational Mechanics and Analysis, vol. 183, no. 2, pp. 215–239, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  8. H. Holden and X. Raynaud, “Global conservative solutions of the Camassa-Holm equation—a lagrangian point of view,” Communications in Partial Differential Equations, vol. 32, no. 10, pp. 1511–1549, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. H. Holden and X. Raynaud, “Global conservative multipeakon solutions of the Camassa-Holm equation,” Journal of Hyperbolic Differential Equations, vol. 4, no. 1, pp. 39–64, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  10. A. Bressan and A. Constantin, “Global dissipative solutions of the Camassa-Holm equation,” Analysis and Applications, vol. 5, no. 1, pp. 1–27, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. Holden and X. Raynaud, “Dissipative solutions for the Camassa-Holm equation,” Discrete and Continuous Dynamical Systems, vol. 24, no. 4, pp. 1047–1112, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. H. Holden and X. Raynaud, “Global dissipative multipeakon solutions of the Camassa-Holm equation,” Communications in Partial Differential Equations, vol. 33, no. 11, pp. 2040–2063, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. A. Constantin and R. Ivanov, “On an integrable two-component Camassa–Holm shallow water system,” Physics Letters A, vol. 372, no. 48, pp. 7129–7132, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. M. Chen, S.-Q. Liu, and Y. Zhang, “A two-component generalization of the Camassa-Holm equation and its solutions,” Letters in Mathematical Physics, vol. 75, no. 1, pp. 1–15, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  15. G. Falqui, “On a Camassa–Holm type equation with two dependent variables,” Journal of Physics A: Mathematical and General, vol. 39, no. 2, pp. 327–342, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  16. P. Olver and P. Rosenau, “Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support,” Physical Review E, vol. 53, Article ID 1900, 1996. View at Publisher · View at Google Scholar · View at MathSciNet
  17. R. S. Johnson, “Camassa–Holm, Korteweg–de Vries and related models for water waves,” Journal of Fluid Mechanics, vol. 455, pp. 63–82, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. G. Gui and Y. Liu, “On the Cauchy problem for the two-component Camassa–Holm system,” Mathematische Zeitschrift, vol. 268, no. 1-2, pp. 45–66, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. J. Escher, O. Lechtenfeld, and Z. Y. Yin, “Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation,” Discrete and Continuous Dynamical Systems, vol. 19, no. 3, pp. 493–513, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. Y. Wang and Y. Song, “Global conservative and multipeakon conservative solutions for the two-component Camassa-Holm system,” Boundary Value Problems, vol. 2013, article 165, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  21. C. Guan and Z. Y. Yin, “Global existence and blow-up phenomena for an integrable two-component Camassa–Holm shallow water system,” Journal of Differential Equations, vol. 248, no. 8, pp. 2003–2014, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  22. G. Gui and Y. Liu, “On the global existence and wave-breaking criteria for the two-component Camassa–Holm system,” Journal of Functional Analysis, vol. 258, no. 12, pp. 4251–4278, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  23. C. Guan and Z. Yin, “Global weak solutions for a two-component Camassa–Holm shallow water system,” Journal of Functional Analysis, vol. 260, no. 4, pp. 1132–1154, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. Y. Wang and Y. Song, “On the global existence of dissipative solutions for the modified coupled Camassa–Holm system,” Soft Computing, vol. 17, no. 1, pp. 2007–2019, 2013. View at Publisher · View at Google Scholar
  25. Z. Shen, Y. Wang, H. Karimi, and Y. Song, “On the multipeakon dissipative behavior of the modiied coupled Camassa-Holm model for shallow water system,” Mathematical Problems in Engineering, vol. 2013, Article ID 107450, 11 pages, 2013. View at Publisher · View at Google Scholar