Global Smooth Solution of Nonlinear Schr$\ddot{o}$dinger Equation

Yongqian Han

Published 2013-07-04, updated 2013-07-31Version 2

The spatially periodic initial problem and Cauchy problem for nonlinear Schr$\ddot{o}$dinger 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.