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

Cuccagna, Scipio; Visciglia, Nicola

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

Cuccagna, Scipio ¹ ; Visciglia, Nicola 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, Tome 347 (2009) no. 5-6, pp. 243-247

Abstract (VO)
Résumé

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) .

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)

.

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 :

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

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

[4] Ginibre, J.; Velo, G. Time decay of finite energy solutions of the nonlinear Klein Gordon and Schrödinger equations, Ann. Inst. H. Poincaré A, Volume 43 (1985), pp. 399-442

[5] Keel, M.; Tao, T. Endpoint Strichartz estimates, Amer. J. Math., Volume 120 (1998) no. 5, pp. 955-980

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