Global well-posedness for the higher order non-linear Schr\"odinger equation on modulations spaces

TL;DR

This paper proves the global well-posedness of a higher order nonlinear Schrödinger equation in modulation spaces for initial data with certain regularity, extending the understanding of solution behavior in these function spaces.

Contribution

It establishes the global well-posedness of the higher order nonlinear Schrödinger equation in modulation spaces for low regularity initial data, using advanced analytical techniques.

Findings

Global well-posedness in modulation spaces for s ≥ 1/4 and p ≥ 2

Extension of solution theory to lower regularity initial data

Application of ideas from Killip, Visan, Zhang, Oh, Wang

Abstract

We consider the initial value problem (IVP) associated to a higher order nonlinear Schr\"odinger (h-NLS) equation $ \partial_{t}u+ia \partial^{2}_{x}u+ b\partial^{3}_{x}u+ic_1|u|^{2}u+c_2 |u|^{2}\partial_{x}u=0, \quad x,t \in \mathbb{R}, $ for given data in the modulation space $M_s^{2,p}(\mathbb{R})$. Using ideias of Killip, Visan, Zhang, Oh, Wang, we prove that the IVP associated to the h-NLS equation is globally well-posed in the modulation spaces $M^{s,p}$ for $s\geq\frac14$ and $p\geq2$.