Scattering for small energy solutions of NLS with periodic potential in 1D

Scipio Cuccagna; Nicola Visciglia

doi:10.1016/j.crma.2009.01.028

Partial Differential Equations

Scattering for small energy solutions of NLS with periodic potential in 1D
[Scattering pour NLS avec des données petites et potentiel périodique]

Présenté par : Jean-Michel Bony

Scipio Cuccagna ¹ ; Nicola Visciglia ²

¹ DISMI, Università di Modena e Reggio Emilia, via Amendola 2, Padiglione Morselli, 42100 Reggio Emilia, Italy
² Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, Italy

Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 243-247.

Résumé
Abstract

On étudie l'existence de l'opérateur de scattering pour le problème de Cauchy suivant :

i \partial_{t} u + H u = μ {| u |}^{p - 1} u, (t, x) \in R \times R, u (0) = u_{0} \in H^{1} (R)

où

H \equiv - \partial_{x}^{2} + V (x)

,

V : R \to R

est un potentiel périodique régulier,

μ \in R \ {0}

,

p ⩾ 7

et

u_{0}

est une petite donnée dans l'espace

H^{1} (R)

.

Given $H \equiv - \partial_{x}^{2} + V (x)$ with $V : R \to R$ a smooth periodic potential, for $μ \in R \ {0}$ and $p ⩾ 7$ , we prove scattering for small solutions to

i \partial_{t} u + H u = μ {| u |}^{p - 1} u, (t, x) \in R \times R, u (0) = u_{0} \in H^{1} (R) .

Reçu le : 2008-08-21
Accepté le : 2009-01-27
Publié le : 2009-02-20

DOI : 10.1016/j.crma.2009.01.028

Affiliations des auteurs :

Scipio Cuccagna ¹ ; Nicola Visciglia ²

¹ DISMI, Università di Modena e Reggio Emilia, via Amendola 2, Padiglione Morselli, 42100 Reggio Emilia, Italy
² Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56100 Pisa, Italy

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     author = {Scipio Cuccagna and Nicola Visciglia},
     title = {Scattering for small energy solutions of {NLS} with periodic potential in {1D}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {243--247},
     publisher = {Elsevier},
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     number = {5-6},
     year = {2009},
     doi = {10.1016/j.crma.2009.01.028},
     language = {en},
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Scipio Cuccagna; Nicola Visciglia. Scattering for small energy solutions of NLS with periodic potential in 1D. Comptes Rendus. Mathématique, Volume 347 (2009) no. 5-6, pp. 243-247. doi : 10.1016/j.crma.2009.01.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.01.028/

Bibliographie
Cité par

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[3] S. Cuccagna; N. Visciglia On asymptotic stability of ground states of NLS with a finite bands periodic potential in 1D | arXiv

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Luigi Forcella; Nicola Visciglia Double scattering channels for 1 $D$ NLS in the energy space and its generalization to higher dimensions, Journal of Differential Equations, Volume 264 (2018) no. 2, pp. 929-958 | DOI:10.1016/j.jde.2017.09.027 | Zbl:1397.35270
L. Miguel Rodrigues Linear asymptotic stability and modulation behavior near periodic waves of the Korteweg-de Vries equation, Journal of Functional Analysis, Volume 274 (2018) no. 9, pp. 2553-2605 | DOI:10.1016/j.jfa.2018.02.004 | Zbl:1392.35267
Scipio Cuccagna; Nicola Visciglia On asymptotic stability of ground states of NLS with a finite bands periodic potential in 1D, Transactions of the American Mathematical Society, Volume 363 (2011) no. 5, pp. 2357-2391 | DOI:10.1090/s0002-9947-2010-05046-9 | Zbl:1298.35191
Scipio Cuccagna; Mirko Tarulli On Asymptotic Stability of Standing Waves of Discrete Schrödinger Equation in $Z$ , SIAM Journal on Mathematical Analysis, Volume 41 (2009) no. 3, p. 861 | DOI:10.1137/080732821

Cité par 4 documents. Sources : Crossref, zbMATH

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