Instability for Semi-classical Schrödinger Equations

Geometric Optics and Instability for Semi-classical Schrödinger Equations

We prove some instability phenomena for semi-classical (linear or) nonlinear Schrödinger equations. For some perturbations of the data, we show that for very small times, we can neglect the Laplacian, and the mechanism is the same as for the corresponding ordinary differential equation. Our approach allows smaller perturbations of the data, where the instability occurs for times such that the p...

A note on Berestycki-Cazenave's classical instability result for nonlinear Schrödinger equations

In this note we give an alternative, shorter proof of the classical result of Berestycki and Cazenave on the instability by blow-up for the standing waves of some nonlinear Schrödinger equations.

Orbital stability of bound states of nonlinear Schrödinger equations with linear and nonlinear lattices

We study the orbital stability and instability of single-spike bound states of critical semi-classical nonlinear Schrödinger equations (NLS) with linear and nonlinear lattices. These equations may model an inhomogeneous Bose-Einstein condensate and an optical beam in a nonlinear lattice. When the linear lattice is switched off, we derive the asymptotic expansion formulas and obtain necessary co...

Semi-classical Measures for Inhomogeneous Schrödinger Equations on Tori

Semiclassical Limit of the Nonlinear Schrödinger-Poisson Equation with Subcritical Initial Data

We study the semi-classical limit of the nonlinear Schrödinger-Poisson (NLSP) equation for initial data of the WKB type. The semi-classical limit in this case is realized in terms of a density-velocity pair governed by the Euler-Poisson equations. Recently we have shown in [ELT, Indiana Univ. Math. J., 50 (2001), 109–157], that the isotropic Euler-Poisson equations admit a critical threshold ph...