A lower index bilinear estimate for the quadratic Schr\"odinger equation and application for its half line problem

TL;DR

This paper establishes local well-posedness for the quadratic Schrödinger equation on a half-line with low regularity initial and boundary data, using a boundary integral operator approach.

Contribution

It introduces a new lower index bilinear estimate and applies it to improve well-posedness results for the half-line problem.

Findings

Proves local well-posedness under low regularity assumptions.

Develops a new bilinear estimate for the quadratic Schrödinger equation.

Utilizes boundary integral operator method for the analysis.

Abstract

We prove the local well-posedness of the initial boundary value problem for the nonlinear quadratic Schr\"odinger equation under low initial-boundary regularity assumption via the boundary integral operator method introduced by Bona-Sun-Zhang (Trans. Amer. Math. Soc., 354(2):427-490, 2002).