no. 3
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

Boutet de Monvel, Anne1 ; Shepelsky, Dmitry2
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, Tome 352 (2014) no. 3, pp. 189-195.

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.01.001
Boutet de Monvel, Anne 1 ; Shepelsky, Dmitry 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 
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Boutet de Monvel, Anne; Shepelsky, Dmitry. The Ostrovsky–Vakhnenko equation: A Riemann–Hilbert approach. Comptes Rendus. Mathématique, Tome 352 (2014) no. 3, pp. 189-195. doi : 10.1016/j.crma.2014.01.001. http://www.numdam.org/articles/10.1016/j.crma.2014.01.001/

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