Orbital Stability of Optical Solitons in 2d

TL;DR

This paper establishes the orbital stability of optical solitons in a 2D Schrödinger-Poisson system modeling light propagation in liquid crystals, including existence and stability results for ground states.

Contribution

It provides the first stability proof for ground states in a nonlocal 2D Schrödinger-Poisson model with non-scaling nonlinearities, and demonstrates existence of ground states for all frequencies in (0,1).

Findings

Proved orbital stability of ground states.

Established existence of ground states for all frequencies in (0,1).

Developed coercivity bounds for the second derivative of the action.

Abstract

We present a stability result for ground states of a Schr\"odinger-Poisson system in $(2+1)$ dimension, modelling the propagation of a light beam through a liquid crystal with nonlocal nonlinear response. The core of the proof is a coercivity bound on the second derivative of the action, where non scaling nonlinearities and the coupled system present the major difficulties. In addition we prove existence of a ground state with frequency $\sigma$ for any $\sigma \in (0,1)$ as a minimal point over an appropriate Nehari manifold.