[L'équation de Camassa–Holm sur la droite par la méthode de Riemann–Hilbert]
Nous étudions par la méthode de « Riemann–Hilbert » le problème de Cauchy pour l'équation de Camassa–Holm (CH) sur la droite : . Nous obtenons que : (i) pour tout , 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 , ce train de solitons tend vers un train de « peakons », solutions lisses par morceaux de l'équation CH pour .
We present a Riemann–Hilbert problem formalism for the initial value problem for the Camassa–Holm equation on the line (CH). We show that: (i) for all , 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 .
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@article{CRMATH_2006__343_10_627_0,
author = {Boutet de Monvel, Anne and Shepelsky, Dmitry},
title = {Riemann{\textendash}Hilbert approach for the {Camassa{\textendash}Holm} equation on the line},
journal = {Comptes Rendus. Math\'ematique},
pages = {627--632},
publisher = {Elsevier},
volume = {343},
number = {10},
year = {2006},
doi = {10.1016/j.crma.2006.10.014},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2006.10.014/}
}
TY - JOUR AU - Boutet de Monvel, Anne AU - Shepelsky, Dmitry TI - Riemann–Hilbert approach for the Camassa–Holm equation on the line JO - Comptes Rendus. Mathématique PY - 2006 SP - 627 EP - 632 VL - 343 IS - 10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2006.10.014/ DO - 10.1016/j.crma.2006.10.014 LA - en ID - CRMATH_2006__343_10_627_0 ER -
%0 Journal Article %A Boutet de Monvel, Anne %A Shepelsky, Dmitry %T Riemann–Hilbert approach for the Camassa–Holm equation on the line %J Comptes Rendus. Mathématique %D 2006 %P 627-632 %V 343 %N 10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2006.10.014/ %R 10.1016/j.crma.2006.10.014 %G en %F CRMATH_2006__343_10_627_0
Boutet de Monvel, Anne; Shepelsky, Dmitry. Riemann–Hilbert approach for the Camassa–Holm equation on the line. Comptes Rendus. Mathématique, Tome 343 (2006) no. 10, pp. 627-632. doi : 10.1016/j.crma.2006.10.014. http://www.numdam.org/articles/10.1016/j.crma.2006.10.014/
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