Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials

Benedetta  Noris; Hugo  Tavares; Gianmaria Verzini

doi:10.3934/dcds.2015.35.6085

Article Contents

2015, Volume 35, Issue 12: 6085-6112. Doi: 10.3934/dcds.2015.35.6085

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Stable solitary waves with prescribed $L^2$-mass for the cubic Schrödinger system with trapping potentials

1.
INdAM-COFUND Marie Curie Fellow, Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin, 45 avenue des Etats-Unis, 78035 Versailles Cédex
2.
Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa
3.
Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, p.za Leonardo da Vinci 32, 20133 Milano

Received: April 30, 2014

Published: November 2015

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Abstract

For the cubic Schrödinger system with trapping potentials in $\mathbb{R}^N$, $N\leq3$, or in bounded domains, we investigate the existence and the orbital stability of standing waves having components with prescribed $L^2$-mass. We provide a variational characterization of such solutions, which gives information on the stability through a condition of Grillakis-Shatah-Strauss type. As an application, we show existence of conditionally orbitally stable solitary waves when: a) the masses are small, for almost every scattering lengths, and b) in the defocusing, weakly interacting case, for any masses.

Keywords:

Cooperative and competitive elliptic systems,
Gross-Pitaevskii systems,
constrained critical points,
Ambrosetti-Prodi type problem,
orbital stability.

Mathematics Subject Classification: Primary: 35C08, 35Q55, 35J50; Secondary: 35B40.

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