A 2D Schrodinger equation with time-oscillating exponential nonlinearity

Abstract

This paper deals with the 2-D Schrödinger equation with time-oscillating exponential nonlinearity $i{\partial }_{t}u+\Delta u=\theta (\omega t)(e^{4\pi |u{|}^2}-1)$, where $\theta$ is a periodic $C^1$-function. We prove that for a class of initial data $u_0\in H^1({\mathbb{R}}^2)$, the solution $u_{\omega }$ converges, as $|\omega |$ tends to infinity to the solution $U$ of the limiting equation $i{\partial }_{t}U+\Delta U=I(\theta )(e^{4\pi |U{|}^2}-1)$ with the same initial data, where $I(\theta )$ is the average of $\theta$.

Submitted