Well-posedness for a system of quadratic derivative nonlinear Schr\"odinger equations with low regularity periodic initial data

TL;DR

This paper proves the well-posedness of a quadratic derivative nonlinear Schrödinger system with low regularity periodic initial data, extending previous nonperiodic results to the periodic setting at critical regularity for dimensions three and higher.

Contribution

It establishes the well-posedness of the system for periodic initial data at the scaling critical regularity, which was previously known only for the nonperiodic case.

Findings

Well-posedness at critical regularity for $d\geq 3$ in the periodic case.

Extension of nonperiodic results to periodic boundary conditions.

Conditions on Laplacian coefficients ensuring well-posedness.

Abstract

We consider the Cauchy problem of a system of quadratic derivative nonlinear Schr\"odinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. For the nonperiodic case, the author proved the small data global well-posedness and the scattering at the scaling critical regularity for $d\geq 2$ when the coefficients of Laplacian satisfy some condition. In the present paper, we prove the well-posedness of the system for the periodic case. In particular, well-posedness is proved at the scaling critical regularity for $d\geq 3$ under some condition for the coefficients of Laplacian.