Stabilité asymptotique des ondes solitaires de l'équation de Benjamin–Bona–Mahony

Khaled El Dika

doi:10.1016/j.crma.2003.10.003

Équations aux dérivées partielles

Stabilité asymptotique des ondes solitaires de l'équation de Benjamin–Bona–Mahony
[Asymptotic stability of solitary waves for the Benjamin–Bona–Mahony equation]

Presented by: Pierre-Louis Lions

Khaled El Dika ¹

¹ Laboratoire de mathématiques, Université Paris-Sud, 91405 Orsay, France

Comptes Rendus. Mathématique, Volume 337 (2003) no. 10, pp. 649-652.

Résumé (Original language)
Abstract

Nous montrons la stabilité asymptotique dans $H^{1} (ℝ)$ de la famille des ondes solitaires de l'équation de Benjamin–Bona–Mahony. Nous obtenons ce résultat en montrant un théorème de rigidité du flot autour des ondes solitaires (un théorème de type Liouville).

We prove the asymptotic stability in $H^{1} (ℝ)$ of the family of solitary waves for the Benjamin–Bona–Mahony equation. The proof is based on a Liouville type theorem on the flow around the solitary waves.

Received: 2003-06-24
Accepted: 2003-10-07
Published online: 2003-11-05

DOI: 10.1016/j.crma.2003.10.003

Author's affiliations:

Khaled El Dika ¹

¹ Laboratoire de mathématiques, Université Paris-Sud, 91405 Orsay, France

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Khaled El Dika. Stabilité asymptotique des ondes solitaires de l'équation de Benjamin–Bona–Mahony. Comptes Rendus. Mathématique, Volume 337 (2003) no. 10, pp. 649-652. doi : 10.1016/j.crma.2003.10.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.10.003/

References
Cited by

[1] T.B. Benjamin; J.L. Bona; J.J. Mahony Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, Volume 272 (1972) no. 1220, pp. 47-78

[2] J.L. Bona; A. Soyeur On the stability of solitary-waves solutions of model equations for long waves, J. Nonlinear Sci., Volume 4 (1994) no. 5, pp. 449-470

[3] K. El Dika, Asymptotic stability of solitary waves for the Benjamin–Bona–Mahony equation, en préparation

[4] Y. Martel; F. Merle A Liouville theorem for the critical generalized Korteweg–de Vries equation, J. Math. Pures Appl. (9), Volume 79 (2000) no. 4, pp. 339-425

[5] Y. Martel; F. Merle Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Rational Mech. Anal., Volume 157 (2001) no. 3, pp. 219-254

[6] J.R. Miller; M.I. Weinstein Asymptotic stability of solitary waves for the regularized long-wave equation, Comm. Pure Appl. Math., Volume 49 (1996) no. 4, pp. 399-441

[7] M.I. Weinstein Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differential Equations, Volume 12 (1987) no. 10, pp. 1133-1173

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