[L'équation de Camassa–Holm sur la demi-droite avec condition aux limites linéarisable]
Nous considérons un problème aux limites pour l'équation de Camassa–Holm sur la demi-droite avec condition de Dirichlet homogène au bord . Nous montrons que, comme dans le cas du problème de Cauchy sur la droite, la solution s'exprime, sous forme paramétrique, en termes de la solution d'un problème de Riemann–Hilbert auxiliaire, entièrement déterminé par des fonctions spectrales associées aux seules données initiales. Cela permet d'appliquer la méthode de plus grande descente non linéaire et d'obtenir ainsi le comportement asymptotique de la solution pour les grandes valeurs du temps. Cette analyse met en évidence trois secteurs du quadrant , où la solution a des comportements asymptotiques de types différents.
We present a Riemann–Hilbert problem formalism for the initial boundary value problem for the Camassa–Holm equation on the half-line with homogeneous Dirichlet boundary condition at . We show that, similarly to the problem on the whole line, the solution of this problem can be obtained in parametric form via the solution of a Riemann–Hilbert problem determined only by the initial data via associated spectral functions. This allows us to apply the nonlinear steepest descent method and to describe the large-time asymptotics of the solution. There are three sectors of the quarter plane , where the asymptotic behavior is qualitatively different.
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@article{CRMATH_2010__348_13-14_775_0,
author = {Boutet de Monvel, Anne and Shepelsky, Dmitry},
title = {The {Camassa{\textendash}Holm} equation on the half-line with linearizable boundary condition},
journal = {Comptes Rendus. Math\'ematique},
pages = {775--780},
publisher = {Elsevier},
volume = {348},
number = {13-14},
year = {2010},
doi = {10.1016/j.crma.2010.05.002},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2010.05.002/}
}
TY - JOUR AU - Boutet de Monvel, Anne AU - Shepelsky, Dmitry TI - The Camassa–Holm equation on the half-line with linearizable boundary condition JO - Comptes Rendus. Mathématique PY - 2010 SP - 775 EP - 780 VL - 348 IS - 13-14 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2010.05.002/ DO - 10.1016/j.crma.2010.05.002 LA - en ID - CRMATH_2010__348_13-14_775_0 ER -
%0 Journal Article %A Boutet de Monvel, Anne %A Shepelsky, Dmitry %T The Camassa–Holm equation on the half-line with linearizable boundary condition %J Comptes Rendus. Mathématique %D 2010 %P 775-780 %V 348 %N 13-14 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2010.05.002/ %R 10.1016/j.crma.2010.05.002 %G en %F CRMATH_2010__348_13-14_775_0
Boutet de Monvel, Anne; Shepelsky, Dmitry. The Camassa–Holm equation on the half-line with linearizable boundary condition. Comptes Rendus. Mathématique, Tome 348 (2010) no. 13-14, pp. 775-780. doi : 10.1016/j.crma.2010.05.002. http://www.numdam.org/articles/10.1016/j.crma.2010.05.002/
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