Ill-posedness for a system of quadratic nonlinear Schrödinger equations in two dimensions

This paper concerns the ill-posedness issue for a system of quadratic nonlinear Schrödinger equations in two dimensions. From previous studies for the large time behavior of the solution to the system, one may expect that the critical regularity to show the well-posedness depends on the dispersion coefficients. To prove the ill-posedness, we show the failure of the uniform continuity of the data-solution map for the mass resonance case and show the norm inflation for other cases.