The Ostrovsky–Vakhnenko equation: A Riemann–Hilbert approach

Anne Boutet de Monvel; Dmitry Shepelsky

doi:10.1016/j.crma.2014.01.001

Partial differential equations/Mathematical physics

The Ostrovsky–Vakhnenko equation: A Riemann–Hilbert approach
[L'équation d'Ostrovsky–Vakhnenko : Une approche de type Riemann–Hilbert]

Présenté par : Jean-Michel Bony

Anne Boutet de Monvel ¹ ; Dmitry Shepelsky ²

¹ Institut de mathématiques de Jussieu–PRG, Université Denis-Diderot (Paris-7), case 7012, bât. Sophie-Germain, 75205 Paris cedex 13, France
² Verkin Institute for Low Temperature Physics and Engineering, 47 Lenin Avenue, 61103 Kharkiv, Ukraine

Comptes Rendus. Mathématique, Volume 352 (2014) no. 3, pp. 189-195.

Résumé
Abstract

Nous présentons une étude par diffusion inverse de l'équation (différentiée) d'Ostrovsky–Vakhnenko :

u_{t x x} - 3 u_{x} + 3 u_{x} u_{x x} + u u_{x x x} = 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.

We present an inverse scattering transform approach for the (differentiated) Ostrovsky–Vakhnenko equation:

u_{t x x} - 3 u_{x} + 3 u_{x} u_{x x} + u u_{x x x} = 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.

Reçu le : 2013-08-09
Accepté le : 2014-01-06
Publié le : 2014-02-05

DOI : 10.1016/j.crma.2014.01.001

Affiliations des auteurs :

Anne Boutet de Monvel ¹ ; Dmitry Shepelsky ²

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     journal = {Comptes Rendus. Math\'ematique},
     pages = {189--195},
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Anne Boutet de Monvel; Dmitry Shepelsky. The Ostrovsky–Vakhnenko equation: A Riemann–Hilbert approach. Comptes Rendus. Mathématique, Volume 352 (2014) no. 3, pp. 189-195. doi : 10.1016/j.crma.2014.01.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.01.001/

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