The Camassa–Holm equation on the half-line with linearizable boundary condition

Anne Boutet de Monvel; Dmitry Shepelsky

doi:10.1016/j.crma.2010.05.002

Partial Differential Equations/Mathematical Physics

The Camassa–Holm equation on the half-line with linearizable boundary condition
[L'équation de Camassa–Holm sur la demi-droite avec condition aux limites linéarisable]

Présenté par : Gilles Lebeau

Anne Boutet de Monvel ¹ ; Dmitry Shepelsky ²

¹ Institut de mathématiques de Jussieu, université Paris Diderot Paris 7, 175, rue du Chevaleret, 75013 Paris, France
² Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine

Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 775-780.

Résumé
Abstract

Nous considérons un problème aux limites pour l'équation de Camassa–Holm sur la demi-droite $x > 0$ avec condition de Dirichlet homogène au bord $x = 0$ . Nous montrons que, comme dans le cas du problème de Cauchy sur la droite, la solution $u (x, t)$ 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 $x > 0$ , $t > 0$ 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 $x > 0$ with homogeneous Dirichlet boundary condition at $x = 0$ . 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 $x > 0$ , $t > 0$ where the asymptotic behavior is qualitatively different.

Reçu le : 2010-01-19
Accepté le : 2010-05-27
Publié le : 2010-06-17

DOI : 10.1016/j.crma.2010.05.002

Affiliations des auteurs :

Anne Boutet de Monvel ¹ ; Dmitry Shepelsky ²

¹ Institut de mathématiques de Jussieu, université Paris Diderot Paris 7, 175, rue du Chevaleret, 75013 Paris, France
² Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine

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     title = {The {Camassa{\textendash}Holm} equation on the half-line with linearizable boundary condition},
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Anne Boutet de Monvel; Dmitry Shepelsky. The Camassa–Holm equation on the half-line with linearizable boundary condition. Comptes Rendus. Mathématique, Volume 348 (2010) no. 13-14, pp. 775-780. doi : 10.1016/j.crma.2010.05.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.05.002/

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Anne Boutet de Monvel; Dmitry Shepelsky Initial-boundary value problem for the Camassa-Holm equation with linearizable boundary condition, Letters in Mathematical Physics, Volume 96 (2011) no. 1-3, pp. 123-141 | DOI:10.1007/s11005-010-0457-6 | Zbl:1230.37089
Anne Boutet de Monvel; Dmitry Shepelsky The Camassa-Holm equation on the half-line: a Riemann-Hilbert approach, The Journal of Geometric Analysis, Volume 18 (2008) no. 2, pp. 285-323 | DOI:10.1007/s12220-008-9014-2 | Zbl:1157.37334

Cité par 2 documents. Sources : zbMATH

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