Global Well-Posedness of 2D Non-Focusing Schr\"odinger Equations via Rigorous Modulation Approximation

TL;DR

This paper proves global well-posedness for certain 2D nonlinear Schrödinger equations with large initial data by rigorously justifying modulation approximations, showing solutions grow at most polynomially over time.

Contribution

It introduces a rigorous modulation approximation method to establish global well-posedness for a class of 2D NLS with large Sobolev data, including indefinite signature cases.

Findings

Solutions grow at most polynomially in time

Global well-posedness established for subcritical Sobolev data

Applicable to NLS with elliptic and indefinite signatures

Abstract

We consider the long time well-posedness of the Cauchy problem with large Sobolev data for a class of nonlinear Schr\"odinger equations (NLS) on $\mathbb{R}^2$ with power nonlinearities of arbitrary odd degree. Specifically, the method in this paper applies to those NLS equations having either elliptic signature with a defocusing nonlinearity, or else having an indefinite signature. By rigorously justifying that these equations govern the modulation of wave packet-like solutions to an artificially constructed equation with an advantageous structure, we show that a priori every subcritical inhomogeneous Sobolev norm of the solution increases at most polynomially in time. Global well-posedness follows by a standard application of the subcritical local theory.