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

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≥ 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≥ 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.