Riemann–Hilbert approach for the Camassa–Holm equation on the line

Anne Boutet de Monvel; Dmitry Shepelsky

doi:10.1016/j.crma.2006.10.014

Partial Differential Equations/Mathematical Physics

Riemann–Hilbert approach for the Camassa–Holm equation on the line
[L'équation de Camassa–Holm sur la droite par la méthode de Riemann–Hilbert]

Présenté par : Paul Malliavin

Anne Boutet de Monvel ¹ ; Dmitry Shepelsky ²

¹ Institut de mathématiques de Jussieu, case 7012, université Paris 7, 2, place Jussieu, 75251 Paris cedex 05, France
² Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine

Comptes Rendus. Mathématique, Volume 343 (2006) no. 10, pp. 627-632.

Résumé
Abstract

Nous étudions par la méthode de « Riemann–Hilbert » le problème de Cauchy pour l'équation de Camassa–Holm (CH) sur la droite : $u_{t} - u_{t x x} + 2 ω u_{x} + 3 u u_{x} = 2 u_{x} u_{x x} + u u_{x x x}$ . Nous obtenons que : (i) pour tout $ω > 0$ , la solution du problème de Cauchy s'exprime de façon paramétrique en termes de la solution d'un problème de Riemann–Hilbert associé ; (ii) cette solution a pour asymptotique, pour t grand, un train de solitons lisses ; (iii) pour $ω \to 0$ , ce train de solitons tend vers un train de « peakons », solutions lisses par morceaux de l'équation CH pour $ω = 0$ .

We present a Riemann–Hilbert problem formalism for the initial value problem for the Camassa–Holm equation $u_{t} - u_{t x x} + 2 ω u_{x} + 3 u u_{x} = 2 u_{x} u_{x x} + u u_{x x x}$ on the line (CH). We show that: (i) for all $ω > 0$ , the solution of this problem can be obtained in a parametric form via the solution of some associated Riemann–Hilbert problem; (ii) for large time, it develops into a train of smooth solitons; (iii) for small ω, this soliton train is close to a train of peakons, which are piecewise smooth solutions of the CH equation for $ω = 0$ .

Reçu le : 2006-08-31
Accepté le : 2006-10-12
Publié le : 2006-11-13

DOI : 10.1016/j.crma.2006.10.014

Affiliations des auteurs :

Anne Boutet de Monvel ¹ ; Dmitry Shepelsky ²

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     author = {Anne Boutet de Monvel and Dmitry Shepelsky},
     title = {Riemann{\textendash}Hilbert approach for the {Camassa{\textendash}Holm} equation on the line},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {627--632},
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     language = {en},
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Anne Boutet de Monvel; Dmitry Shepelsky. Riemann–Hilbert approach for the Camassa–Holm equation on the line. Comptes Rendus. Mathématique, Volume 343 (2006) no. 10, pp. 627-632. doi : 10.1016/j.crma.2006.10.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.10.014/

Bibliographie
Cité par

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