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

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

itu+Hu=μ|u|p1u,(t,x)R×R,u(0)=u0H1(R)
Hx2+V(x), V:RR est un potentiel périodique régulier, μR\{0}, p7 et u0 est une petite donnée dans l'espace H1(R).

Given Hx2+V(x) with V:RR a smooth periodic potential, for μR\{0} and p7, we prove scattering for small solutions to

itu+Hu=μ|u|p1u,(t,x)R×R,u(0)=u0H1(R).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.01.028

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},
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     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/

[1] T. Cazenave Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, vol. 10, New York University, Courant Institute of Mathematical Sciences, New York, 2003

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

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

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

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