Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2015, Article ID 925715, 7 pages
http://dx.doi.org/10.1155/2015/925715
Research Article

Orbital Stability of Solitary Traveling Waves of Moderate Amplitude

Department of Mathematics, Foshan University, Foshan, Guangdong 528000, China

Received 2 December 2014; Accepted 24 January 2015

Academic Editor: Stephen C. Anco

Copyright © 2015 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. 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 MathSciNet · View at Scopus
  2. A. Geyer, “Solitary traveling water waves of moderate amplitude,” Journal of Nonlinear Mathematical Physics, vol. 19, supplement 1, pp. 104–115, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. 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 MathSciNet · View at Scopus
  4. 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
  5. A. Constantin, “On the scattering problem for the Camassa-Holm equation,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 457, pp. 953–970, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  6. 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 · View at MathSciNet · View at Scopus
  7. A. Boutet de Monvel, A. Kostenko, D. Shepelsky, and G. Teschl, “Long-time asymptotics for the Camassa-Holm equation,” SIAM Journal on Mathematical Analysis, vol. 41, no. 4, pp. 1559–1588, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. K. El Dika and L. Molinet, “Exponential decay of H1-localized solutions and stability of the train of N solitary waves for the Camassa-Holm equation,” Philosophical Transactions of the Royal Society of London Series A, vol. 365, no. 1858, pp. 2313–2331, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. Z. R. 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
  10. W. L. Zhang, “General expressions of peaked travelling wave solutions of CH-γ and CH equations,” Science in China Series A: Mathematics, vol. 47, no. 6, pp. 862–873, 2004. View at Google Scholar
  11. 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 MathSciNet · View at Scopus
  12. A. Constantin and J. Escher, “Analyticity of periodic traveling free surface water waves with vorticity,” Annals of Mathematics, vol. 173, no. 1, pp. 559–568, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. 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
  14. A. Constantin and L. Molinet, “Global weak solutions for a shallow water equation,” Communications in Mathematical Physics, vol. 211, no. 1, pp. 45–61, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. 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 MathSciNet
  16. 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 Google Scholar · View at MathSciNet · View at Scopus
  17. 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
  18. Z. Y. Ouyang, S. Zheng, and Z. R. Liu, “Orbital stability of peakons with nonvanishing boundary for CH and CH-γequations,” Physics Letters A, vol. 372, no. 47, pp. 7046–7050, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. T. B. Benjamin, “The stability of solitary waves,” Proceedings of the Royal Society—London Series A, vol. 328, pp. 153–183, 1972. View at Publisher · View at Google Scholar · View at MathSciNet
  20. K. El Dika and L. Molinet, “Exponential decay of H1-localized solutions and stability of the train of N solitary waves for the Camassa-Holm equation,” Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 365, no. 1858, pp. 2313–2331, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. K. El Dika and L. Molinet, “Stability of multipeakons,” Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire, vol. 26, no. 4, pp. 1517–1532, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. 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
  23. N. DurukMutlubas and A. Geyer, “Orbital stability of solitary waves of moderate amplitude in shallow water,” Journal of Differential Equations, vol. 255, no. 2, pp. 254–263, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. N. D. Mutlubaş, “On the Cauchy problem for a model equation for shallow water waves of moderate amplitude,” Nonlinear Analysis: Real World Applications, vol. 14, no. 5, pp. 2022–2026, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus