The nonlinear Schrödinger equation ground states on product spaces

Terracini, Susanna; Tzvetkov, Nikolay; Visciglia, Nicola

doi:10.2140/apde.2014.7.73

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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

DOI: 10.2140/apde.2014.7.73

Abstract

We study the nature of the nonlinear Schrödinger equation ground states on the product spaces $ℝ^{n} \times M^{k}$ , 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

Mathematical Subject Classification 2010

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

Vol. 7, No. 1, 2014