Global Smooth Solution of Nonlinear Schr\"odinger Equation

TL;DR

This paper proves the existence and uniqueness of smooth, global solutions for a broad class of nonlinear Schrödinger equations with various initial data conditions, including critical and supercritical cases.

Contribution

It establishes the global well-posedness of nonlinear Schrödinger equations with infinite smooth initial data and bounded Fourier support, covering critical and supercritical regimes.

Findings

Global smooth solutions exist for infinite smooth initial data.

Solutions are unique and globally well-posed under certain spectral conditions.

The results apply to both defocusing and focusing nonlinear Schrödinger equations.

Abstract

The spatially periodic initial problem and Cauchy problem for nonlinear Schr\"odinger equations are considered. The existence and uniqueness of global solution with infinite smooth initial data $u_0$, i.e. $u_0,\;|u_0|^{2p}u_0\in H^{\infty}$, are established. Moreover these two problem with initial data $u_0\in H^m$ are globally well-posed provided the Fourier frequency of $u_0$ is contained in a bounded compact set. The equations studied here cover $L^2$ and $\dot{H}^1$ critical and supercritical, defocusing and focusing nonlinear Schr$\ddot{o}$dinger equations.