Geodesic
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, Fethi1 ; Malchiodi, Andrea2 ; Montenegro, Marcelo3
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.

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We study the nonlinear Schrödinger equation −ε2Δψ+V(x)ψ=|ψ|p−1ψ on a compact manifold or on Rn, 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 −ε2Δψ+V(x)ψ=|ψ|p−1ψ sur une variété compacte ou sur Rn, 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.

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DOI : 10.1016/j.crma.2007.11.008

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 
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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. http://geodesic.mathdoc.fr/articles/10.1016/j.crma.2007.11.008/

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