Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 52020, 36 pages
http://dx.doi.org/10.1155/2007/52020
Research Article

Existence and Orbital Stability of Cnoidal Waves for a 1D Boussinesq Equation

1Departamento de Matemática, Instituto de Matemática, Estatística e Computaçäo Científica, (IMECC), UNICAMP, CP 6065, Campinas, Säo Paulo CEP 13083-970, Brazil
2Departamento de Matemáticas, Universidad del Valle, Cali A. A. 25360, Colombia

Received 11 May 2006; Revised 1 October 2006; Accepted 7 February 2007

Academic Editor: Vladimir Mityushev

Copyright © 2007 Jaime Angulo and Jose R. Quintero. 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. J. Angulo, Existence and Stability of Solitary Wave Solutions to Nonlinear Dispersive Evolution Equations, Publicaçőes Matemáticas do IMPA, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil, 2003, 24 Colóquio Brasileiro de Matemática. View at Zentralblatt MATH · View at MathSciNet
  2. J. Angulo, “Stability of cnoidal waves to Hirota-Satsuma systems,” Matemática Contemporânea, vol. 27, pp. 189–223, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J. Angulo, J. Bona, and M. Scialom, “Stability of cnoidal waves,” Advances in Differential Equations, vol. 11, pp. 1321–1374, 2006. View at Google Scholar
  4. 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 Zentralblatt MATH · View at MathSciNet
  5. T. B. Benjamin, “The stability of solitary waves,” Proceedings of the Royal Society. London. Series A, vol. 328, pp. 153–183, 1972. View at Google Scholar · View at MathSciNet
  6. J. L. Bona, “On the stability theory of solitary waves,” Proceedings of the Royal Society. London. Series A, vol. 344, no. 1638, pp. 363–374, 1975. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. I. Weinstein, “Lyapunov stability of ground states of nonlinear dispersive evolution equations,” Communications on Pure and Applied Mathematics, vol. 39, no. 1, pp. 51–67, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J. H. Maddocks and R. L. Sachs, “Constrained variational principles and stability in Hamiltonian systems,” in Hamiltonian Dynamical Systems (Cincinnati, OH, 1992), H. S. Dumas, K. R. Meyer, and D. S. Schmidt, Eds., vol. 63 of IMA Volumes in Mathematics and Its Applications, pp. 231–264, Springer, New York, NY, USA, 1995. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. R. Quintero, “Nonlinear stability of a one-dimensional Boussinesq equation,” Journal of Dynamics and Differential Equations, vol. 15, no. 1, pp. 125–142, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. Grillakis, J. Shatah, and W. Strauss, “Stability theory of solitary waves in the presence of symmetry. II,” Journal of Functional Analysis, vol. 94, no. 2, pp. 308–348, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J. Boussinesq, “Théorie de l'intumescence liquide, applelée onde solitaire ou de translation, se propageant dans un canal rectangulaire,” Comptes Rendus de l'Academie des Sciences, vol. 72, pp. 755–759, 1871. View at Google Scholar
  12. J. Boussinesq, “Théorie des ondes te des remous qui se propageant le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond,” Journal de Mathématiques Pures et Appliquées, vol. 17, pp. 55–108, 1872. View at Google Scholar
  13. D. J. Korteweg and G. de Vries, “On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,” Philosophical Magazine Series 5, vol. 39, pp. 422–443, 1895. View at Google Scholar
  14. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, vol. 67 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, NY, USA, 2nd edition, 1971. View at Zentralblatt MATH · View at MathSciNet
  15. W. Magnus and S. Winkler, Hill's Equation, vol. 2 of Tracts in Pure and Appl. Math., John Wiley & Sons, New York, NY, USA, 1976.
  16. E. L. Ince, “The periodic Lamé functions,” Proceedings of the Royal Society of Edinburgh, vol. 60, pp. 47–63, 1940. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983. View at Zentralblatt MATH · View at MathSciNet
  18. J. P. Albert and J. L. Bona, “Total positivity and the stability of internal waves in stratified fluids of finite depth,” IMA Journal of Applied Mathematics, vol. 46, no. 1-2, pp. 1–19, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet