Global well-posedness and scattering for the fourth order nonlinear Schr\"{o}dinger equations with small data

TL;DR

This paper proves global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small initial data across various dimensions, using advanced harmonic analysis techniques.

Contribution

It establishes the existence of scattering operators and global solutions for small data in Besov spaces, extending previous results to higher-order Schrödinger equations.

Findings

Global well-posedness for $n extgreater 3$ in Besov spaces

Existence of scattering operators for small data

New estimates for fourth order Schrödinger semi-groups

Abstract

For $n\geq 3$, we study the Cauchy problem for the fourth order nonlinear Schr\"{o}dinger equations, for which the existence of the scattering operators and the global well-posedness of solutions with small data in Besov spaces $B^{s}_{2,1}(\R^n)$ are obtained. In one spatial dimension, we get the global well-posedness result with small data in the critical homogeneous Besov spaces $\dot{B}^{s}_{2,1}$. As a by-product, the existence of the scattering operators with small data is also obtained. In order to show these results, the global version of the estimates for the maximal functions and the local smoothing effects on the fourth order Schr\"{o}dinger semi-groups are established.