A 2D Schrodinger equation with time-oscillating exponential nonlinearity

Abstract

This paper deals with the 2-D Schr\"odinger equation with time-oscillating exponential nonlinearity $i\partial_t u+\Delta u= \theta(\omega t)\big(e^{4\pi|u|^2}-1\big)$, 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 $i\partial_t U+\Delta U= I(\theta)\big(e^{4\pi|U|^2}-1\big)$ with the same initial data, where $I(\theta)$ is the average of $\theta$.