Solutions to the nonlinear Schrödinger equation carrying momentum along a curve

Fethi Mahmoudi; Andrea Malchiodi; Marcelo Montenegro

doi:10.1016/j.crma.2007.11.008

Partial Differential Equations

Solutions to the nonlinear Schrödinger equation carrying momentum along a curve

Presented by: Haïm Brezis

Fethi Mahmoudi ¹; Andrea Malchiodi ²; Marcelo Montenegro 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, Volume 346 (2008) no. 1-2, pp. 33-38.

Abstract
Résumé

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.

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.

Received: 2007-08-01
Accepted: 2007-10-28
Published online: 2007-12-03

DOI: 10.1016/j.crma.2007.11.008

Author's affiliations:

Fethi Mahmoudi ¹; Andrea Malchiodi ²; Marcelo Montenegro 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

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     author = {Fethi Mahmoudi and Andrea Malchiodi and Marcelo Montenegro},
     title = {Solutions to the nonlinear {Schr\"odinger} equation carrying momentum along a curve},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {33--38},
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     number = {1-2},
     year = {2008},
     doi = {10.1016/j.crma.2007.11.008},
     language = {en},
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Fethi Mahmoudi; Andrea Malchiodi; Marcelo Montenegro. Solutions to the nonlinear Schrödinger equation carrying momentum along a curve. Comptes Rendus. Mathématique, Volume 346 (2008) no. 1-2, pp. 33-38. doi : 10.1016/j.crma.2007.11.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.11.008/

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