The NLS ground states on product spaces

TL;DR

This paper investigates the properties of ground states for the nonlinear Schrödinger equation on product spaces, revealing how these states depend on mass and the geometry of the manifold, with implications for stability.

Contribution

It establishes the behavior of ground states on product spaces, showing their similarity to Euclidean ground states at small mass and their dependence on the manifold at larger mass, along with stability analysis.

Findings

Ground states match Euclidean states at small mass

Ground states depend on the manifold at large mass

Addresses stability of solutions

Abstract

We study the nature of the Nonlinear Schr\"odinger 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 transform the variational analysis to dynamical stability results.