Local well‐posedness for a system of quadratic nonlinear Schrödinger equations in one or two dimensions

In this article, the local well‐posedness of Cauchy's problem is explored for a system of quadratic nonlinear Schrödinger equations in the space L^p( R ⁿ). In a special case of mass resonant 2 × 2 system, it is well known that this problem is well posed in H^s(s≥0) and ill posed in H^s(s < 0) in two‐space dimensions. By translation on a linear semigroup, we show that the general system becomes locally well posed in L^p( R ²) for 1 < p < 2, for which p can arbitrarily be close to the scaling limit p_c=1. In one‐dimensional case, we show that the problem is locally well posed in L¹( R ); moreover, it has a measure valued solution if the initial data are a Dirac function. Copyright © 2016 John Wiley & Sons, Ltd.