Orbital stability of bound states of nonlinear Schrodinger equations with linear and nonlinear optical lattices

TL;DR

This paper investigates the orbital stability of bound states in nonlinear Schrödinger equations with optical lattices, providing new conditions and formulas for stability analysis relevant to Bose-Einstein condensates.

Contribution

It introduces comprehensive mathematical theorems for orbital stability of bound states in NLS equations with linear and nonlinear optical lattices, including asymptotic expansions and stability criteria.

Findings

Derived asymptotic expansion formulas for stability analysis.

Established necessary conditions for stability and instability.

Developed general theorems applicable to various optical lattice configurations.

Abstract

We study the orbital stability and instability of single-spike bound states of semiclassical nonlinear Schrodinger (NLS) equations with critical exponent, linear and nonlinear optical lattices (OLs). These equations may model two-dimensional Bose-Einstein condensates in linear and nonlinear OLs. When linear OLs are switched off, we derive the asymptotic expansion formulas and obtain necessary conditions for the orbital stability and instability of single-spike bound states, respectively. When linear OLs are turned on, we consider three different conditions of linear and nonlinear OLs to develop mathematical theorems which are most general on the orbital stability problem.