The nonlinear Quadratic Interactions of the Schr\"odinger type on the half-line

TL;DR

This paper establishes local well-posedness for coupled Schrödinger equations with quadratic nonlinearities on the half-line, using advanced bilinear estimates and Bourgain space techniques to handle low regularity data.

Contribution

It introduces new bilinear estimates in Bourgain spaces tailored for the Schrödinger system with quadratic nonlinearities on the half-line, advancing the understanding of low regularity solutions.

Findings

Proved local well-posedness for low regularity Sobolev data.

Developed new bilinear estimates in Bourgain spaces with $b<1/2$.

Analyzed the dispersion relation's role in regularity requirements.

Abstract

In this work we study the initial boundary value problem associated with the coupled Schr\"odinger equations {with quadratic nonlinearities, that appears in nonlinear optics}, on the half-line. We obtain local well-posedness for data {in Sobolev spaces} with low regularity, by using a forcing problem on the full line with a presence of a forcing term in order to apply the Fourier restriction method of Bourgain. The crucial point in this work is the new bilinear estimates on the classical Bourgain spaces $X^{s,b}$ with $b<\frac12$, jointly with bilinear estimates in adapted Bourgain spaces that will used to treat the traces of nonlinear part of the solution. Here the understanding of the dispersion relation is the key point in these estimates, where the set of regularity depends strongly of the constant $a$ measures the scaling-diffraction magnitude indices.