Vol. 7, No. 1, 2014

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The nonlinear Schrödinger equation ground states on product spaces

Susanna Terracini, Nikolay Tzvetkov and Nicola Visciglia

Vol. 7 (2014), No. 1, 73–96
Abstract

We study the nature of the nonlinear Schrödinger equation ground states on the product spaces ${ℝ}^{n}×{M}^{k}$, where ${M}^{k}$ is a compact Riemannian manifold. We prove that for small ${L}^{2}$ masses the ground states coincide with the corresponding ${ℝ}^{n}$ ground states. We also prove that above a critical mass the ground states have nontrivial ${M}^{k}$ dependence. Finally, we address the Cauchy problem issue, which transforms the variational analysis into dynamical stability results.

Keywords
NLS, stability, stability of solitons, rigidity, ground states
Primary: 35Q55
Secondary: 37K45
Milestones
Received: 2 May 2012
Accepted: 21 May 2013
Published: 7 May 2014
Authors
 Susanna Terracini Dipartimento di Matematica G. Peano Università di Torino Via Carlo Alberto 10 I-10123 Torino Italy Nikolay Tzvetkov Département de Mathématiques Université de Cergy Pontoise 2, Avenue A. Chauvin 95302 Cergy-Pontoise France Institut Universitaire de France Nicola Visciglia Dipartimento di Matematica Università Degli Studi di Pisa Largo Bruno Pontecorvo 5 I-56127 Pisa Italy