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

We study the nature of the nonlinear Schrödinger equation ground states on the product spaces n × Mk, where Mk is a compact Riemannian manifold. We prove that for small L2 masses the ground states coincide with the corresponding n ground states. We also prove that above a critical mass the ground states have nontrivial Mk dependence. Finally, we address the Cauchy problem issue, which transforms the variational analysis into dynamical stability results.

NLS, stability, stability of solitons, rigidity, ground states
Mathematical Subject Classification 2010
Primary: 35Q55
Secondary: 37K45
Received: 2 May 2012
Accepted: 21 May 2013
Published: 7 May 2014
Susanna Terracini
Dipartimento di Matematica G. Peano
Università di Torino
Via Carlo Alberto 10
I-10123 Torino
Nikolay Tzvetkov
Département de Mathématiques
Université de Cergy Pontoise
2, Avenue A. Chauvin
95302 Cergy-Pontoise
Institut Universitaire de France
Nicola Visciglia
Dipartimento di Matematica
Università Degli Studi di Pisa
Largo Bruno Pontecorvo 5
I-56127 Pisa