Scattering in the weighted $L^2$-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity

Abstract

We investigate the defocusing inhomogeneous nonlinear Schrödinger equation $$i{\partial }_{t}u+\Delta u=|x{|}^{-b}\left({\mathrm{e}}^{\alpha |u{|}^2}-1-\alpha |u{|}^2\right)u,u(0)=u_0,x\in {\mathbb{R}}^2,$$ with $0<b<1$ and $\alpha =2\pi (2-b)$. First we show the decay of global solutions by assuming that the initial data $u_0$ belongs to the weighted space $\Sigma ({\mathbb{R}}^2)=\{u\in H^1({\mathbb{R}}^2):|x|u\in L^2({\mathbb{R}}^2)\}$. Then we combine the local theory with the decay estimate to obtain scattering in $\Sigma$ when the Hamiltonian is below the value $\frac{2}{(1+b)(2-b)}$.

23 pages