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

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     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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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. https://www.numdam.org/articles/10.1016/j.crma.2007.11.008/

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