Homogenization for the cubic nonlinear Schr\"odinger equation on $\mathbb R^2$

TL;DR

This paper investigates the behavior of the inhomogeneous mass-critical nonlinear Schr"odinger equation on , establishing conditions for global solutions and homogenization as the inhomogeneity scale becomes large.

Contribution

It provides new sufficient conditions on the inhomogeneity function to guarantee global existence, uniqueness, and homogenization for large inhomogeneity scales.

Findings

Global solutions exist under certain conditions on g for large n.

Homogenization occurs as n approaches infinity.

Conditions on g ensure well-posedness of the equation.

Abstract

We study the defocusing inhomogeneous mass-critical nonlinear Schr\"odinger equation on $\mathbb R^2$ $$i u_t +\Delta u=g(nx) |u|^2 u$$ for initial data in $L^2(\mathbb R^2)$. We obtain sufficient conditions on $g$ to ensure existence and uniqueness of global solutions for $n$ sufficiently large, as well as homogenization.