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Comptes Rendus Mathématique
Volume 352, n° 3
pages 189-195 (mars 2014)
Doi : 10.1016/j.crma.2014.01.001
Received : 9 August 2013 ;  accepted : 6 January 2014
The Ostrovsky–Vakhnenko equation: A Riemann–Hilbert approach
L'équation d'Ostrovsky–Vakhnenko : Une approche de type Riemann–Hilbert
 

Anne Boutet de Monvel a , Dmitry Shepelsky b
a Institut de mathématiques de Jussieu–PRG, Université Denis-Diderot (Paris-7), case 7012, bât. Sophie-Germain, 75205 Paris cedex 13, France 
b Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, 61103 Kharkiv, Ukraine 

Abstract

We present an inverse scattering transform approach for the (differentiated) Ostrovsky–Vakhnenko equation:
utxx−3ux+3uxuxx+uuxxx=0. This equation can also be viewed as the short-wave model for the Degasperis–Procesi equation. The approach is based on an associated Riemann–Hilbert problem, which allows us to give a representation for the classical (smooth) solution of the Cauchy problem, to get the principal term of its long-time asymptotics, and also to find, in a natural way, loop soliton solutions.

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Résumé

Nous présentons une étude par diffusion inverse de l'équation (différentiée) d'Ostrovsky–Vakhnenko :
utxx−3ux+3uxuxx+uuxxx=0. Cette équation peut aussi se voir comme le modèle « ondes courtes » de l'équation de Degasperis–Procesi. Notre approche consiste à se ramener à l'étude d'un problème de Riemann–Hilbert associé. Elle nous permet d'obtenir une représentation de la solution classique (lisse) du problème de Cauchy et de déterminer le terme principal de l'asymptotique à temps grand de cette solution. Elle permet aussi d'obtenir, de façon naturelle, des solutions solitons de type à boucle.

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