ON THE CAUCHY PROBLEM OF A COHERENTLY COUPLED SCHR?DINGER SYSTEM

In this article, we consider the well-posedness of a coherently coupled Schrdinger system with four waves mixing in space dimension n ≤ 4. The Cauchy problem for the cubic system is studied in L~2 for n ≤ 2 and in H~1 for n ≤ 4. We obtain two sharp conditions between global existence and blow up.