On the low-regularity global well-posedness of a system of nonlinear Schrodinger Equation

TL;DR

This paper establishes low-regularity global well-posedness results for a one-dimensional quadratic nonlinear Schrödinger system with a parameter, using advanced decomposition techniques to handle different parameter ranges.

Contribution

It provides new global well-posedness results for the system at low regularity levels, adapting decomposition methods for different parameter regimes.

Findings

Global well-posedness for $rac{1}{2}< ext{α}<1$ in $H^s$ with $s>-rac{1}{4}$.

Global well-posedness for $0< ext{α}<rac{1}{2}$ in $H^s$ with $s>-rac{5}{8}$.

Use of linear-nonlinear and resonance decomposition techniques adapted to parameter ranges.

Abstract

In this article, we study the low-regularity Cauchy problem of a one dimensional quadratic Schrodinger system with coupled parameter $\alpha\in (0, 1)$. When $\frac{1}{2}<\alpha<1$,we prove the global well-posedness in $H^s(\mathbb{R})$ with $s>-\frac{1}{4}$, while for $0<\alpha<\frac{1}{2}$, we obtain global well-posedness in $H^s(\mathbb{R})$ with $s>-\frac{5}{8}$. We have adapted the linear-nonlinear decomposition and resonance decomposition technique in different range of $\alpha$.