Local well-posedness for a system of quadratic derivative nonlinear Schr\"{o}dinger equations

TL;DR

This paper improves the local well-posedness results for a quadratic derivative nonlinear Schrödinger system modeling laser-plasma interactions by employing the short-time Fourier restriction norm method.

Contribution

It advances previous work by establishing better well-posedness results under certain conditions using a novel analytical approach.

Findings

Enhanced well-posedness results for the system.

Application of the short-time Fourier restriction norm method.

Improved understanding of the system's flow map properties.

Abstract

We consider the Cauchy problem for a system of quadratic derivative nonlinear Schr\"odinger equations introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. Under the condition that the flow map fails to be twice differentiable, Hirayama, Kinoshita, and Okamoto (2022) proved the well-posedness by constructing a modified energy and applying the energy method. In the present paper, we improve the well-posedness result under the above condition by using the short-time Fourier restriction norm method.