On the blow-up threshold for weakly coupled nonlinear Schrodinger equations

We study the Cauchy problem for a system of two coupled nonlinear focusing Schrodinger equations arising in nonlinear optics. We discuss when the solutions are global in time or blow-up in finite time. Some results, in dependence of the data of the problem, are proved; in particular we prove, for suitable values of the parameters, that the blow-up threshold ( if the nonlinearity has the critical growth) is a universal constant.