Well-posedness and ill-posedness for a system of periodic quadratic derivative nonlinear Schr\"odinger equations

TL;DR

This paper establishes well-posedness and ill-posedness results for a system of quadratic derivative nonlinear Schrödinger equations in the periodic setting, extending previous nonperiodic results and identifying critical regularity thresholds.

Contribution

It proves the well-posedness of the system at the scaling critical regularity for dimensions three and higher in the periodic case, under specific coefficient conditions, and also demonstrates ill-posedness in certain cases.

Findings

Well-posedness at critical regularity for d≥3 in periodic setting

Ill-posedness results for certain coefficient conditions

Extension of nonperiodic results to periodic case

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 setting, the authors proved some well-posedness results, which contain the scaling critical case for $d\geq 2$. In the present paper, we prove the well-posedness of this system for the periodic setting. In particular, well-posedness is proved at the scaling critical regularity for $d\geq 3$ under some conditions for the coefficients of the Laplacian. We also prove some ill-posedness results. As long as we use an iteration argument, our well-posedness results are optimal except for some critical cases.