On the cauchy problem of a coherently coupled schrödinger system

In this article, we consider the well-posedness of a coherently coupled Schrödinger system with four waves mixing in space dimension n ≤ 4. The Cauchy problem for the cubic system is studied in L2 for n ≤ 2 and in H1 for n ≤ 4. We obtain two sharp conditions between global existence and blow up.