Quadratic Nonlinear Derivative Schr\"odiger Equations - Part 1

TL;DR

This paper establishes local well-posedness for quadratic derivative nonlinear Schrödinger equations in 2+1 dimensions with low regularity, small initial data, and spherical symmetry, advancing understanding of these complex PDEs.

Contribution

It provides the first local well-posedness results for quadratic derivative nonlinear Schrödinger equations with low regularity data in 2+1 dimensions, considering derivative nonlinearities.

Findings

Proved local well-posedness up to the scaling for small symmetric initial data.

Extended the theory to include derivatives in the nonlinearity.

Demonstrated well-posedness in low regularity Sobolev spaces.

Abstract

In this paper we consider the local well-posedness theory for the quadratic nonlinear Schr\"odinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a local well-posedness result up to the scaling for small initial data with some spherical symmetry structure.