[L'équation KdV sur un intervalle borné par la méthode de Riemann–Hilbert]
Nous étudions un problème aux limites pour l'équation KdV sur un intervalle borné : l'étude est faite en termes d'un problème de Riemann–Hilbert singulier dans le k-plan complexe pour une fonction matricielle qui dépend de façon explicite de variables d'espace–temps. Pour un ensemble particulier de données de Cauchy ainsi que de valeurs aux limites, nous donnons les « fonctions spectrales » qui rendent la solution du problème de Riemann–Hilbert unique. A partir de cette solution on obtient une expression intégrale de la solution du problème aux limites pour l'équation KdV.
The initial-boundary value problem for the KdV equation on a finite interval is analyzed in terms of a singular Riemann–Hilbert problem for a matrix-valued function in the complex k-plane which depends explicitly on the space–time variables. For an appropriate set of initial and boundary data, we derive the k-dependent “spectral functions” which guarantee the uniqueness of Riemann–Hilbert problem's solution. The latter determines a solution of the initial-boundary value problem for KdV equation, for which an integral representation is given.
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@article{CRMATH_2009__347_5-6_261_0,
author = {Hitzazis, Iasonas and Tsoubelis, Dimitri},
title = {Riemann{\textendash}Hilbert formulation for the {KdV} equation on a finite interval},
journal = {Comptes Rendus. Math\'ematique},
pages = {261--266},
publisher = {Elsevier},
volume = {347},
number = {5-6},
year = {2009},
doi = {10.1016/j.crma.2009.01.012},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2009.01.012/}
}
TY - JOUR AU - Hitzazis, Iasonas AU - Tsoubelis, Dimitri TI - Riemann–Hilbert formulation for the KdV equation on a finite interval JO - Comptes Rendus. Mathématique PY - 2009 SP - 261 EP - 266 VL - 347 IS - 5-6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2009.01.012/ DO - 10.1016/j.crma.2009.01.012 LA - en ID - CRMATH_2009__347_5-6_261_0 ER -
%0 Journal Article %A Hitzazis, Iasonas %A Tsoubelis, Dimitri %T Riemann–Hilbert formulation for the KdV equation on a finite interval %J Comptes Rendus. Mathématique %D 2009 %P 261-266 %V 347 %N 5-6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2009.01.012/ %R 10.1016/j.crma.2009.01.012 %G en %F CRMATH_2009__347_5-6_261_0
Hitzazis, Iasonas; Tsoubelis, Dimitri. Riemann–Hilbert formulation for the KdV equation on a finite interval. Comptes Rendus. Mathématique, Tome 347 (2009) no. 5-6, pp. 261-266. doi : 10.1016/j.crma.2009.01.012. http://www.numdam.org/articles/10.1016/j.crma.2009.01.012/
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