Scattering for the two-dimensional NLS with (full) exponential nonlinearity

TL;DR

This paper proves global well-posedness and scattering for a 2D nonlinear Schrödinger equation with exponential nonlinearity, using perturbative methods based on mass-critical NLS estimates.

Contribution

It introduces a perturbative approach to handle the exponential nonlinearity by relating it to the mass-critical NLS, extending known techniques to a new class of nonlinearities.

Findings

Established global well-posedness for the equation.

Proved scattering and spacetime bounds for solutions.

Connected exponential nonlinearity analysis with mass-critical NLS techniques.

Abstract

We obtain global well-posedness, scattering, and global $L_t^4H_{x}^{1,4}$ spacetime bounds for energy-space solutions to the energy-subcritical nonlinear Schr\"odinger equation \[iu_t+\Delta u=u(e^{4\pi |u|^2}-1)\] in two spatial dimensions. Our approach is perturbative; we view our problem as a perturbation of the mass-critical NLS to employ the techniques of Tao--Visan--Zhang. This permits us to combine the known spacetime estimates for mass-critical NLS proved by Dodson and the work of Ibrahim--Majdoub--Masmoudi--Nakanishi on a related problem to prove corresponding spacetime estimates which imply scattering.