The NLS ground states on product spaces

Abstract

We study the nature of the Nonlinear Schrödinger equation ground states on the product spaces ${\mathbb{R}}^n\times M^k$, where $M^k$ is a compact Riemannian manifold. We prove that for small $L^2$ masses the ground states coincide with the corresponding ${\mathbb{R}}^n$ 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 transform the variational analysis to dynamical stability results.

24 pages

References in corpus (2)

• Global well-posedness of the energy-critical defocusing NLS on $\mathbb{R}\times {\mathbb{T}}^3$
• Stability and instability of the KdV solitary wave under the KP-I flow

Cited by in corpus (5)

• Existence and Stability of standing waves for supercritical NLS with a Partial Confinement
• Global Well-posedness and scattering for fourth-order Schrödinger equations on waveguide manifolds
• Scattering for the nonlinear Schrodinger equation with a general one-dimensional confinement
• Periodic traveling waves for nonlinear Schrödinger equations with non-zero conditions at infinity in ${\mathbb{R}}^2$
• Uniqueness and orbital stability of standing waves for the nonlinear Schrodinger equation with a partial confinement