#### 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