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

Boutet de Monvel, Anne; Shepelsky, Dmitry

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 ; Shepelsky, Dmitry

Annales de l'Institut Fourier, Tome 59 (2009), p. 3015-3056 / Harvested from Numdam

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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.

Publié le : 2009-01-01

MR 2649345 | Zbl 1191.35245

DOI : https://doi.org/10.5802/aif.2514
Classification: 35Q53, 37K10, 30E25, 35Q15, 37K15, 35B40
Mots clés: équation de Camassa–Holm, asymptotique, problème aux limites, problème de Riemann–Hilbert

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     author = {Boutet de Monvel, Anne and Shepelsky, Dmitry},
     title = {Long time asymptotics of the Camassa--Holm equation on the half-line},
     journal = {Annales de l'Institut Fourier},
     volume = {59},
     year = {2009},
     pages = {3015-3056},
     doi = {10.5802/aif.2514},
     zbl = {1191.35245},
     mrnumber = {2649345},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2009__59_7_3015_0}
}

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) pp. 3015-3056. doi : 10.5802/aif.2514. http://gdmltest.u-ga.fr/item/AIF_2009__59_7_3015_0/

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