Global Well-posedness for the Fourth-order Nonlinear Schrodinger Equation

TL;DR

This paper proves the global well-posedness of the one-dimensional fourth-order nonlinear Schrödinger equation in modulation spaces, using advanced functional analysis tools and a-priori estimates based on perturbation determinants.

Contribution

It introduces a novel approach to establish global well-posedness in modulation spaces for this equation, leveraging the power series expansion of the perturbation determinant.

Findings

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

Use of $U^p-V^p$ spaces and bilinear estimates

A-priori estimates via perturbation determinants

Abstract

The local and global well-posedness for the one dimensional fourth-order nonlinear Schr\"odinger equation are established in the modulation space $M^{s}_{2,q}$ for $s\geq \frac12$ and $2\leq q <\infty$. The local result is based on the $U^p-V^p$ spaces and crucial bilinear estimates. The key ingredient to obtain the global well-posedness is that we achieve a-priori estimates of the solution in modulation spaces by utilizing the power series expansion of the perturbation determinant introduced by Killip-Visan-Zhang for completely integrable PDEs.