Instability of an integrable nonlocal NLS

François Genoud

doi:10.1016/j.crma.2017.01.018

Partial differential equations

Instability of an integrable nonlocal NLS
[De l'instabilité d'une équation NLS non locale intégrable]

Présenté par : Haïm Brézis

François Genoud ¹

¹ Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands

Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 299-303.

Résumé
Abstract

Nous discutons dans cette note la dynamique globale d'une équation NLS intégrable non locale sur $R$ , qui a été étudiée récemment par des méthodes de systèmes intégrables. Nous démontrons deux résultats qui contrastent fortement avec le cas de l'équation NLS focalisante cubique locale. Premièrement, il existe des solutions qui explosent en temps fini, avec condition initiale arbitrairement petite dans $H^{s} (R)$ , pour tout $s ⩾ 0$ . Par ailleurs, les solitons de l'équation NLS locale, qui sont aussi solutions de l'équation non locale, sont instables par explosion pour cette dernière.

In this note, we discuss the global dynamics of an integrable nonlocal NLS on $R$ , which has been the object of a recent investigation by integrable systems methods. We prove two results that are in striking contrast with the case of the local cubic focusing NLS. First, finite-time blow-up solutions exist with arbitrarily small initial data in $H^{s} (R)$ , for any $s ⩾ 0$ . On the other hand, the solitons of the local NLS, which are also solutions to the nonlocal equation, are unstable by blow-up for the latter.

Reçu le : 2016-10-09
Accepté le : 2017-01-26
Publié le : 2017-02-09

DOI : 10.1016/j.crma.2017.01.018

Affiliations des auteurs :

François Genoud ¹

¹ Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands

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     title = {Instability of an integrable nonlocal {NLS}},
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     doi = {10.1016/j.crma.2017.01.018},
     language = {en},
}

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François Genoud. Instability of an integrable nonlocal NLS. Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 299-303. doi : 10.1016/j.crma.2017.01.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.01.018/

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