Solutions to the nonlinear Schrödinger equation carrying momentum along a curve

Mahmoudi, Fethi; Malchiodi, Andrea; Montenegro, Marcelo

doi:10.1016/j.crma.2007.11.008

Partial Differential Equations

Solutions to the nonlinear Schrödinger equation carrying momentum along a curve
[Solutions d'une équation de Schrödinger non linéaire portant un moment le long d'une courbe]

Présenté par : Haïm Brezis

Mahmoudi, Fethi ¹ ; Malchiodi, Andrea ² ; Montenegro, Marcelo ³

¹ Département de Matématiques, Faculté des sciences de Tunis, Campus Universitaire 2092 Tunis, Tunisia
² SISSA, Sector of Mathematical Analysis, Via Beirut 2-4, 34014 Trieste, Italy
³ Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Caixa Postal 6065, CEP 13083-970, Campinas, SP, Brazil

Comptes Rendus. Mathématique, Tome 346 (2008) no. 1-2, pp. 33-38.

Résumé
Abstract

On étudie l'équation de Schrödinger non linéaire $- ε^{2} Δ ψ + V (x) ψ = {| ψ |}^{p - 1} ψ$ sur une variété compacte ou sur $R^{n}$ , où V est un potentiel positif, régulier et $p > 1$ . Lorsque ε tend vers zéro, on montre l'existence de solutions à valeurs complexes qui se concentrent le long d'une courbe fermée et dont la phase est hautement oscillante, portant un moment quantique le long de l'ensemble limite.

We study the nonlinear Schrödinger equation $- ε^{2} Δ ψ + V (x) ψ = {| ψ |}^{p - 1} ψ$ on a compact manifold or on $R^{n}$ , where V is a positive potential and $p > 1$ . As ε tends to zero, we prove existence of complex-valued solutions which concentrate along closed curves and whose phase is highly oscillatory, carrying quantum-mechanical momentum along the limit set.

Reçu le : 2007-08-01
Accepté le : 2007-10-28
Publié le : 2007-12-03

DOI : 10.1016/j.crma.2007.11.008

Affiliations des auteurs :

Mahmoudi, Fethi ¹ ; Malchiodi, Andrea ² ; Montenegro, Marcelo ³

@article{CRMATH_2008__346_1-2_33_0,
     author = {Mahmoudi, Fethi and Malchiodi, Andrea and Montenegro, Marcelo},
     title = {Solutions to the nonlinear {Schr\"odinger} equation carrying momentum along a curve},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {33--38},
     publisher = {Elsevier},
     volume = {346},
     number = {1-2},
     year = {2008},
     doi = {10.1016/j.crma.2007.11.008},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.11.008/}
}

TY  - JOUR
AU  - Mahmoudi, Fethi
AU  - Malchiodi, Andrea
AU  - Montenegro, Marcelo
TI  - Solutions to the nonlinear Schrödinger equation carrying momentum along a curve
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 33
EP  - 38
VL  - 346
IS  - 1-2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2007.11.008/
DO  - 10.1016/j.crma.2007.11.008
LA  - en
ID  - CRMATH_2008__346_1-2_33_0
ER  -

%0 Journal Article
%A Mahmoudi, Fethi
%A Malchiodi, Andrea
%A Montenegro, Marcelo
%T Solutions to the nonlinear Schrödinger equation carrying momentum along a curve
%J Comptes Rendus. Mathématique
%D 2008
%P 33-38
%V 346
%N 1-2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2007.11.008/
%R 10.1016/j.crma.2007.11.008
%G en
%F CRMATH_2008__346_1-2_33_0

Mahmoudi, Fethi; Malchiodi, Andrea; Montenegro, Marcelo. Solutions to the nonlinear Schrödinger equation carrying momentum along a curve. Comptes Rendus. Mathématique, Tome 346 (2008) no. 1-2, pp. 33-38. doi : 10.1016/j.crma.2007.11.008. http://www.numdam.org/articles/10.1016/j.crma.2007.11.008/

Bibliographie
Cité par

[1] Ambrosetti, A.; Malchiodi, A. Perturbation Methods and Semilinear Elliptic Problems on $R^{n}$ , Progr. Math., vol. 240, Birkhäuser, 2005

[2] Ambrosetti, A.; Malchiodi, A.; Ni, W.M. Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I, Commun. Math. Phys., Volume 235 (2003), pp. 427-466

[3] D'Aprile, T. On a class of solutions with non vanishing angular momentum for nonlinear Schrödinger equation, Differential Integral Equations, Volume 16 (2003) no. 3, pp. 349-384

[4] Del Pino, M.; Kowalczyk, M.; Wei, J. Concentration at curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., Volume 60 (2007) no. 1, pp. 113-146

[5] Mahmoudi, F.; Malchiodi, A. Concentration at manifolds of arbitrary dimension for a singularly perturbed Neumann problem, Rend. Lincei Mat. Appl., Volume 17 (2003) no. 2006, pp. 279-290

[6] Mahmoudi, F.; Malchiodi, A. Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. Math., Volume 209 (2007), pp. 460-525

[7] F. Mahmoudi, A. Malchiodi, M. Montenegro, Solutions to the nonlinear Schrödinger equation carrying momentum along a curve. Part I: Study of the limit set and approximate solutions, preprint

[8] F. Mahmoudi, A. Malchiodi, Solutions to the nonlinear Schrödinger equation carrying momentum along a curve. Part II: Proof of the existence result, preprint

[9] Malchiodi, A. Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, GAFA, Volume 15 (2005) no. 6, pp. 1162-1222

[10] Malchiodi, A.; Montenegro, M. Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math., Volume 55 (2002) no. 12, pp. 1507-1568

[11] Malchiodi, A.; Montenegro, M. Multidimensional boundary-layers for a singularly perturbed Neumann problem, Duke Math. J., Volume 124 (2004) no. 1, pp. 105-143

[12] Mazzeo, R.; Pacard, F. Foliations by constant mean curvature tubes, Comm. Anal. Geom., Volume 13 (2005) no. 4, pp. 633-670

[13] Ni, W.M. Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., Volume 45 (1998) no. 1, pp. 9-18

[14] Weinstein, A. Nonlinear stabilization of quasimodes, Univ. Hawaii, Honolulu, Hawaii, 1979 (Proc. Sympos. Pure Math.), Volume vol. XXXVI, Amer. Math. Soc., Providence, RI (1980), pp. 301-318 MR0573443 (82d:58016)

Cité par Sources :