Long time asymptotics of the Camassa–Holm equation on the half-line

Boutet de Monvel, Anne; Shepelsky, Dmitry

doi:10.5802/aif.2514

Long time asymptotics of the Camassa–Holm equation on the half-line
[Étude asymptotique de l’équation de Camassa–Holm sur la demi-droite pour de grandes valeurs du temps]

Boutet de Monvel, Anne ¹ ; Shepelsky, Dmitry ²

¹ Université Paris Diderot Paris 7 Institut de Mathématiques de Jussieu Site Chevaleret, Case 7012 75205 Paris Cedex 13 (France)
² Institute B. Verkin Mathematical Division 47 Lenin Avenue 61103 Kharkiv (Ukraine)

Annales de l'Institut Fourier, Tome 59 (2009) no. 7, pp. 3015-3056.

Résumé
Abstract

Nous étudions le comportement asymptotique, pour de grandes valeurs du temps $t$ , de solutions de problèmes aux limites pour l’équation de Camassa–Holm (CH) $u_{t} - u_{t x x} + 2 u_{x} + 3 u u_{x} = 2 u_{x} u_{x x} + u u_{x x x}$ sur la demi-droite $x \geq 0$ . Cet article prolonge nos travaux antérieurs sur les problèmes aux limites pour l’équation de Camassa–Holm, travaux dont la clef est la formulation et l’analyse de problèmes de Riemann–Hilbert associés. Dans le quart de plan espace-temps $x > 0$ , $t > 0$ , nous distinguons des régions où les solutions ont un comportement asymptotique qualitativement différent, et nous calculons pour chacune d’elles le terme principal de l’asymptotique en termes de données spectrales associées aux valeurs initiales et au bord.

We study the long-time behavior of solutions of the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation $u_{t} - u_{t x x} + 2 u_{x} + 3 u u_{x} = 2 u_{x} u_{x x} + u u_{x x x}$ on the half-line $x \geq 0$ . The paper continues our study of IBV problems for the CH equation, the key tool of which is the formulation and analysis of associated Riemann–Hilbert factorization problems. We specify the regions in the quarter space-time plane $x > 0$ , $t > 0$ having qualitatively different asymptotic pictures, and give the main terms of the asymptotics in terms of spectral data associated with the initial and boundary values.

MR Zbl

DOI : 10.5802/aif.2514

Classification : 35Q53, 37K10, 30E25, 35Q15, 37K15, 35B40
Keywords: Camassa–Holm equation, asymptotics, initial-boundary value problem, Riemann–Hilbert problem
Mot clés : équation de Camassa–Holm, asymptotique, problème aux limites, problème de Riemann–Hilbert

Affiliations des auteurs :

Boutet de Monvel, Anne ¹ ; Shepelsky, Dmitry ²

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Boutet de Monvel, Anne; Shepelsky, Dmitry. Long time asymptotics of the Camassa–Holm equation on the half-line. Annales de l'Institut Fourier, Tome 59 (2009) no. 7, pp. 3015-3056. doi : 10.5802/aif.2514. http://www.numdam.org/articles/10.5802/aif.2514/

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