Orbital stability and uniqueness of the ground state for NLS in dimension one

TL;DR

This paper proves the orbital stability and uniqueness of ground state solutions for the one-dimensional nonlinear Schrödinger equation under certain conditions on the non-linear term, including when it is a pure power-type.

Contribution

It establishes orbital stability and non-degeneracy of ground states for NLS in 1D with specific conditions on the non-linear term, including uniqueness for power-type nonlinearities.

Findings

Standing-wave solutions are orbitally stable under given conditions.

Unique positive symmetric ground state exists for power-type nonlinearities.

Ground states are non-degenerate, ensuring stability and uniqueness.

Abstract

We prove that standing-waves solutions to the non-linear Schr\"odinger equation in dimension one whose profiles can be obtained as minima of the energy over the mass, are orbitally stable and non-degenerate, provided the non-linear term $ G $ satisfies a Euler differential inequality. When the non-linear term $ G $ is a combined pure power-type, then there is only one positive, symmetric minimum of prescribed mass.