Global solutions to 3D quadratic nonlinear Schr\"odinger-type equation

TL;DR

This paper establishes global existence and scattering for small initial data solutions to a 3D fractional quadratic nonlinear Schrödinger equation, using a novel combination of normal form and space-time resonance methods.

Contribution

It introduces a new approach by combining normal form and space-time resonance techniques to analyze the 3D fractional quadratic Schrödinger equation.

Findings

Proves global existence for small initial data.

Demonstrates scattering behavior of solutions.

Develops flexible resolution spaces for controlling interactions.

Abstract

We consider the Cauchy problem to the 3D fractional Schr\"odinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty is that we combine the normal form methods and the space-time resonance methods. Using the normal form transform enables us more flexibilities in designing the resolution spaces so that we can control various interactions. It is also convenient for the final data problem.