Équations aux dérivées partielles
Solutions de type multi-soliton des équations de KdV généralisées
[Multi-soliton-type solutions of the generalized KdV equations]
Comptes Rendus. Mathématique, Volume 338 (2004) no. 6, pp. 457-460.

We consider the generalized Korteweg–de Vries equations in the subcritical and critical cases. Let Rj(t,x)=Qcj(xcjtxj) be N soliton solutions of this equation, with corresponding speeds 0<c1<c2<⋯<cN. In this Note, we give a sketch of the proof of the following result. Given {c j },{x j }, there exists one and only one solution ϕ of the generalized KdV equation such that ‖ϕ(t)−∑Rj(t)‖H1→0 as t→+∞. Complete proofs will appear later.

On considère les équations de Korteweg–de Vries généralisées dans les cas sous-critique et critique. Soit R j (t,x)=Q c j (x-c j t-x j )N solutions de type solitons de l'équation, correspondant à des vitesses 0<c1<c2<⋯<cN. Dans cette Note, on donne les idées principales de la démonstration du résultat suivant. Etant donnés {c j },{x j }, il existe une et une seule solution ϕ de l'équation de KdV généralisée telle que ‖ϕ(t)−∑Rj(t)‖H1→0 quand t→+∞. Les preuves complètes seront publiées plus tard.

Received:
Published online:
DOI: 10.1016/j.crma.2003.12.029
Martel, Yvan 1

1 Centre de mathématiques, École polytechnique, 91128 Palaiseau cedex, France
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Martel, Yvan. Solutions de type multi-soliton des équations de KdV généralisées. Comptes Rendus. Mathématique, Volume 338 (2004) no. 6, pp. 457-460. doi : 10.1016/j.crma.2003.12.029. http://www.numdam.org/articles/10.1016/j.crma.2003.12.029/

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