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

Presented by: 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, Volume 347 (2009) no. 5-6, pp. 243-247.

Abstract
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

u_{0}

est une petite donnée dans l'espace

H^{1} (R)

Received: 2008-08-21
Accepted: 2009-01-27
Published online: 2009-02-20

DOI: 10.1016/j.crma.2009.01.028

Author's affiliations:

Cuccagna, Scipio ¹; Visciglia, Nicola ²

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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},
     publisher = {Elsevier},
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     doi = {10.1016/j.crma.2009.01.028},
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     url = {http://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, Volume 347 (2009) no. 5-6, pp. 243-247. doi : 10.1016/j.crma.2009.01.028. http://www.numdam.org/articles/10.1016/j.crma.2009.01.028/

References
Cited by

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[2] Cuccagna, S. Dispersion for Schrödinger equation with periodic potential in 1D, Comm. Partial Differential Equations, Volume 33 (2008) no. 11, pp. 2064-2095

[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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