Non-linear Smoothing for the Periodic Generalized Non-linear Schr\"odinger Equation

TL;DR

This paper investigates the smoothing effects of the periodic non-linear Schr"odinger equation with odd power non-linearity, establishing new regularity results and global attractor properties for the forced and damped cases.

Contribution

It proves that the difference between non-linear and linear evolutions gains smoothness and extends these results globally for forced and damped equations, including existence of global attractors.

Findings

Difference between non-linear and linear evolutions is smoother

Global smoothing results for forced and damped equations

Existence and regularity of global attractors

Abstract

We consider the periodic non-linear Schr\"odinger equation with non-linearity given by $|u|^{p-1}u$ for odd $p > 1$ in dimension $1$. We first establish that the difference between the non-linear evolution and a phase rotation of the the linear evolution is in a smoother space. We then study forced and damped defocusing non-linear Schr\"odinger equations of the above type and establish an analogous smoothing statement that extends globally in time. As a corollary we establish both existence and smootheness for global attractors in the energy space.