The Cauchy Problem for nonlinear Quadratic Interactions of the Schr\"odinger type in one dimensional space

TL;DR

This paper investigates the well-posedness of coupled nonlinear Schrödinger equations with quadratic interactions in one dimension, establishing local and global results in low regularity Sobolev spaces using bilinear estimates and the I-method.

Contribution

It introduces new bilinear estimates for quadratic coupling terms and extends global well-posedness results to low regularity Sobolev spaces in one-dimensional Schrödinger systems.

Findings

Local well-posedness in low regularity Sobolev spaces

Global well-posedness for negative Sobolev indices

Development of new bilinear estimates for coupling terms

Abstract

In this work I study the well-posedness of the Cauchy problem associated with the coupled Schr\"odinger equations {with quadratic nonlinearities}, which appears modeling problems in nonlinear optics. I obtain the local well-posedness for data {in Sobolev spaces} with low regularity. {To obtain} the local theory, I prove new bilinear estimates for the coupling terms of the system in the continuous case. Concerning global results, in the continuous case, I establish the global well-posedness in $H^s(\mathbb{R})\times H^s(\mathbb{R})$, for some negatives indexes $s$. The proof of the global result uses the \textbf{I}-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao.