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Advances in Mathematical Physics
Volume 2015 (2015), Article ID 789269, 6 pages
http://dx.doi.org/10.1155/2015/789269
Research Article

Stability of Negative Solitary Waves for a Generalized Camassa-Holm Equation with Quartic Nonlinearity

1Department of Basic Courses, Guangzhou Maritime Institute, Guangdong 510725, China
2Department of Mathematics, Foshan University, Guangdong 528000, China

Received 6 October 2015; Revised 1 December 2015; Accepted 8 December 2015

Academic Editor: Christian Engstrom

Copyright © 2015 Shan Zheng and Zhengyong Ouyang. 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. A. S. Fokas and B. Fuchssteiner, “Symplectic structures, their Bäcklund transformations and hereditary symmetries,” Physica D: Nonlinear Phenomena, vol. 4, no. 1, pp. 47–66, 1981. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. 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 · View at MathSciNet · View at Scopus
  3. 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 MathSciNet · View at Scopus
  4. C. E. Kenig, G. Ponce, and L. Vega, “Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle,” Communications on Pure and Applied Mathematics, vol. 46, no. 4, pp. 527–620, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. G. B. Whitham, Linear and Nonlinear Waves, John Wiley & Sons, New York, NY, USA, 1980. View at MathSciNet
  6. 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
  7. A. Constantin, “On the Cauchy problem for the periodic Camassa-Holm equation,” Journal of Differential Equations, vol. 141, no. 2, pp. 218–235, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. G. Misiolek, “Classical solutions of the periodic Camassa-Holm equation,” Geometric and Functional Analysis, vol. 12, no. 5, pp. 1080–1104, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. Z. Xin and P. Zhang, “On the weak solutions to a shallow water equation,” Communications on Pure and Applied Mathematics, vol. 53, no. 11, pp. 1411–1433, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. 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
  11. 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
  12. 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. 10–12, pp. 1511–1549, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. 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 MathSciNet · View at Scopus
  14. A. Constantin and W. A. Strauss, “Stability of peakons,” Communications on Pure and Applied Mathematics, vol. 53, no. 5, pp. 603–610, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. A. Constantin and W. A. Strauss, “Stability of the Camassa-Holm solitons,” Journal of Nonlinear Science, vol. 12, no. 4, pp. 415–422, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. Z.-Y. Ouyang, S. Zheng, and Z.-R. Liu, “Orbital stability of peakons with nonvanishing boundary for CH and CH-γ equations,” Physics Letters A, vol. 372, no. 47, pp. 7046–7050, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. 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 · View at MathSciNet · View at Scopus
  18. A. Constantin, “Existence of permanent 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 · View at MathSciNet
  19. A. Constantin and J. Escher, “Global existence and blow-up for a shallow water equation,” Annali della Scuola Normale Superiore di Pisa—Classe di Scienze, vol. 26, no. 2, pp. 303–328, 1998. View at Google Scholar · View at MathSciNet
  20. A. Constantin and J. Escher, “Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,” Communications on Pure and Applied Mathematics, vol. 51, no. 5, pp. 475–504, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. H. P. McKean, “Breakdown of a shallow water equation,” The Asian Journal of Mathematics, vol. 2, no. 4, pp. 867–874, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. J. Yin, L. Tian, and X. Fan, “Stability of negative solitary waves for an integrable modified Camassa-Holm equation,” Journal of Mathematical Physics, vol. 51, Article ID 053515, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. W. G. Rui, B. He, S. L. Xie, and Y. Long, “Application of the integral bifurcation method for solving modified Camassa-Holm and Degasperis-Procesi equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3459–3470, 2009. View at Publisher · View at Google Scholar
  24. Y. Fu, G. L. Gui, Y. Liu, and C. Z. Qu, “On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity,” Journal of Differential Equations, vol. 255, no. 7, pp. 1905–1938, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. Y. Liu, X. Zhu, and J. He, “Factorization technique and new exact solutions for the modified Camassa-Holm and Degasperis-Procesi equations,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1658–1665, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. A.-M. Wazwaz, “Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations,” Physics Letters A, vol. 352, no. 6, pp. 500–504, 2006. View at Publisher · View at Google Scholar
  27. X. Liu, Y. Liu, and C. Qu, “Orbital stability of the train of peakons for an integrable modified Camassa-Holm equation,” Advances in Mathematics, vol. 255, pp. 1–37, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  28. L. X. Tian and J. L. Yin, “New compacton solutions and solitary wave solutions of fully nonlinear generalized Camassa-Holm equations,” Chaos, Solitons and Fractals, vol. 20, no. 2, pp. 289–299, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. A.-M. Wazwaz, “A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions,” Applied Mathematics and Computation, vol. 165, no. 2, pp. 485–501, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. A.-M. Wazwaz, “New compact and noncompact solutions for two variants of a modified Camassa-Holm equation,” Applied Mathematics and Computation, vol. 163, no. 3, pp. 1165–1179, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  31. R. A. Kraenkel and A. Zenchuk, “Two-dimensional integrable generalization of the Camassa-Holm equation,” Physics Letters A, vol. 260, no. 3-4, pp. 218–224, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  32. Z. Liu and T. Qian, “Peakons and their bifurcation in a generalized Camassa-Holm equation,” International Journal of Bifurcation and Chaos, vol. 11, no. 3, article 781, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. T. F. Qian and M. Y. Tang, “Peakons and periodic cusp waves in a generalized Camassa-Holm equation,” Chaos, Solitons and Fractals, vol. 12, no. 7, pp. 1347–1360, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  34. Z. Y. Liu and T. F. Qian, “Peakons of the Camassa-Holm equation,” Applied Mathematical Modelling, vol. 26, no. 3, pp. 473–480, 2002. View at Publisher · View at Google Scholar · View at Scopus
  35. L. Tian and X. Song, “New peaked solitary wave solutions of the generalized Camassa-Holm equation,” Chaos, Solitons and Fractals, vol. 19, no. 3, pp. 621–637, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. S. A. Khuri, “New ansäz for obtaining wave solutions of the generalized Camassa-Holm equation,” Chaos, Solitons and Fractals, vol. 25, no. 3, pp. 705–710, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. Z. Y. Yin, “On the Cauchy problem for the generalized Camassa-Holm equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 2, pp. 460–471, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. O. G. Mustafa, “On the Cauchy problem for a generalized Camassa-Holm equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 64, no. 6, pp. 1382–1399, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  39. J. Yin, Q. Xing, and L. Tian, “Orbital stability and dynamical behaviors of solitary waves for the Camassa-Holm equation with quartic nonlinearity,” Chaos, Solitons & Fractals, vol. 76, pp. 40–46, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  40. T. B. Benjamin, “The stability of solitary waves,” Proceedings of the Royal Society London Series A: Mathematical, Physical and Engineering Sciences, vol. 328, pp. 153–183, 1972. View at Publisher · View at Google Scholar · View at MathSciNet
  41. T. B. Benjamin, J. L. Bona, and J. J. Mahony, “Model equations for long waves in nonlinear dispersive systems,” Philosophical Transactions of the Royal Society of London A, vol. 272, no. 1220, pp. 47–78, 1972. View at Publisher · View at Google Scholar · View at MathSciNet
  42. M. Grillakis, J. Shatah, and W. Strauss, “Stability theory of solitary waves in the presence of symmetry I,” Journal of Functional Analysis, vol. 74, no. 1, pp. 160–197, 1987. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  43. G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, John Wiley & Sons, 1998. View at MathSciNet
  44. N. Dunford and J. T. Schwarz, Linear Operators, Part II: Spectral Theory, Interscience Publishers, John Wiley and Sons, New York, NY, USA, 1963.