Abstract
We will study the existence and stability of periodic travelling-wave solutions of the nonlinear one-dimensional Boussinesq-type equation
Research Article | Open Access
Existence and Orbital Stability of Cnoidal Waves for a 1D Boussinesq Equation
Academic Editor: Vladimir Mityushev
Received11 May 2006
Revised01 Oct 2006
Accepted07 Feb 2007
Published28 Mar 2007
References
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, Colóquio Brasileiro de Matemática.
View at: Zentralblatt MATH | MathSciNetJ. Angulo, “Stability of cnoidal waves to Hirota-Satsuma systems,” Matemática Contemporânea, vol. 27, pp. 189–223, 2004.
View at: Google Scholar | Zentralblatt MATH | MathSciNetJ. Angulo, J. Bona, and M. Scialom, “Stability of cnoidal waves,” Advances in Differential Equations, vol. 11, pp. 1321–1374, 2006.
View at: Google ScholarM. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNetT. 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 | MathSciNetJ. 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 | Zentralblatt MATH | MathSciNetM. 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: Google Scholar | Zentralblatt MATH | MathSciNetJ. 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 | Zentralblatt MATH | MathSciNetJ. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNetM. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNetJ. 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 ScholarJ. 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 ScholarD. 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 ScholarP. 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 | MathSciNetW. Magnus and S. Winkler, Hill's Equation, vol. 2 of Tracts in Pure and Appl. Math., John Wiley & Sons, New York, NY, USA, 1976.
E. L. Ince, “The periodic Lamé functions,” Proceedings of the Royal Society of Edinburgh, vol. 60, pp. 47–63, 1940.
View at: Google Scholar | Zentralblatt MATH | MathSciNetA. 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 | MathSciNetJ. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
Copyright
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.