VARIATIONAL METHOD FOR STUDYING SELF-FOCUSING IN A CLASS OF NONLINEAR SCHRODINGER-EQUATIONS

We discuss the use of Dirac's variational principle as applied to a class of nonlinear Schrodinger equations, i partial derivative psi/partial derivative t + del2psi + g\psi*psi\(kappa)psi = 0, in d spatial dimensions for arbitrary nonlinearity parameter kappa and for Gaussian initial data at t = 0. A Gaussian trial variational wave function is used to obtain qualitative information about blowup at finite times as a function of d and kappa. For the critical case kappad = 2 we obtain an analytic expression for the critical mass necessary for blowup as a function of d which is in good agreement with the results of numerical calculations in one and two dimensions. for the supercritical case kappad > 2 we determine the critical exponents as a function Of kappa and d.