Well-posedness and scattering for fourth order nonlinear Schr\"odinger type equations at the scaling critical regularity

TL;DR

This paper proves global well-posedness and scattering for a class of fourth order nonlinear Schrödinger equations with derivative nonlinearities at the critical regularity, extending results to higher dimensions and polynomial nonlinearities.

Contribution

It establishes the first well-posedness and scattering results for these equations at the critical regularity, including higher dimensions and polynomial derivatives.

Findings

Global well-posedness for 1D derivative quartic NLS

Scattering to free solutions in the critical setting

Extension to higher dimensions and polynomial nonlinearities

Abstract

In the present paper, we consider the Cauchy problem of fourth order nonlinear Schr\"odinger type equations with a derivative nonlinearity. In one dimensional case, we prove that the fourth order nonlinear Schr\"odinger equation with the derivative quartic nonlinearity $\partial _x (\overline{u}^4)$ is the small data global in time well-posed and scattering to a free solution. Furthermore, we show that the same result holds for the $d \ge 2$ and derivative polynomial type nonlinearity, for example $|\nabla | (u^m)$ with $(m-1)d \ge 4$.