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Orbital Stability of Optical Solitons in 2d

Sergio Moroni

TL;DR

The paper studies orbital stability of ground states for a Schrödinger–Poisson system in two dimensions, modeling optical solitons in nonlocal nematic media. It develops a spectral-coercivity framework based on the second variation of the action around ground states, handling symmetry and nonlocal nonlinearities to prove orbital stability for almost every mass and frequency, with a complementary variational stability result via concentration-compactness. Existence of ground states for every frequency in $(0,1)$ is established through a Nehari-manifold variational approach using a modified energy $E_\sigma$, while a detailed decay analysis shows exponential localization of the stationary profiles. Together, these results provide a rigorous foundation for the stability and structure of optical solitons in nonlocal media and extend the variational analysis to coupled Schrödinger–Poisson systems.

Abstract

We present a stability result for ground states of a Schrödinger-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 $σ$ for any $σ\in (0,1)$ as a minimal point over an appropriate Nehari manifold.

Orbital Stability of Optical Solitons in 2d

TL;DR

The paper studies orbital stability of ground states for a Schrödinger–Poisson system in two dimensions, modeling optical solitons in nonlocal nematic media. It develops a spectral-coercivity framework based on the second variation of the action around ground states, handling symmetry and nonlocal nonlinearities to prove orbital stability for almost every mass and frequency, with a complementary variational stability result via concentration-compactness. Existence of ground states for every frequency in is established through a Nehari-manifold variational approach using a modified energy , while a detailed decay analysis shows exponential localization of the stationary profiles. Together, these results provide a rigorous foundation for the stability and structure of optical solitons in nonlocal media and extend the variational analysis to coupled Schrödinger–Poisson systems.

Abstract

We present a stability result for ground states of a Schrödinger-Poisson system in 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 for any as a minimal point over an appropriate Nehari manifold.
Paper Structure (6 sections, 31 theorems, 155 equations)

This paper contains 6 sections, 31 theorems, 155 equations.

Table of Contents

  1. Preliminaries
  2. Stability of ground states
  3. Orbital Stability
  4. Variational Stability
  5. Existence for any $0<\sigma<1$
  6. Decaying rate

Key Result

Theorem 3

Let $a_0$ be such that for any $a>a_0$, there exists a ground state $(v_a, \phi_a)$ over $S_a$. Then for almost every $a>a_0$ the ground state is orbitally stable.

Theorems & Definitions (66)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Proposition 4
  • Theorem 5
  • Remark 6
  • Proposition 7
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • ...and 56 more