Well-posedness and ill-posedness for a system of periodic quadratic derivative nonlinear Schrödinger equations

Abstract

We consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger 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 $d\ge 2$. 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 $d\ge 3$ 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.

52 pages

References in corpus (3)

• On the 2d Zakharov system with L^2 Schrödinger data
• On the well-posedness problem for the derivative nonlinear Schrödinger equation
• Norm inflation for a higher-order nonlinear Schrödinger equation with a derivative on the circle