Energy Scattering for Schr\"{o}dinger Equation with Exponential Nonlinearity in Two Dimensions

TL;DR

This paper proves that the scattering operator is well-defined in the entire energy space for a 2D nonlinear Schrödinger equation with exponential nonlinearity, under specific initial data and energy conditions.

Contribution

It establishes the global well-posedness and scattering theory for the Schrödinger equation with exponential nonlinearity in two dimensions.

Findings

Scattering operator is well-defined in H^1 for initial data with energy ≤ 1.

Results hold for exponential nonlinearity with parameter λ<4π.

Provides a rigorous foundation for scattering in this nonlinear regime.

Abstract

When the spatial dimensions $n$=2, the initial data $u_0\in H^1$ and the Hamiltonian $H(u_0)\leq 1$, we prove that the scattering operator is well-defined in the whole energy space $H^1(\mathbb{R}^2)$ for nonlinear Schr\"{o}dinger equation with exponential nonlinearity $(e^{\lambda|u|^2}-1)u$, where $0<\lambda<4\pi$.