POST-GAUSSIAN VARIATIONAL METHOD FOR THE NONLINEAR SCHRODINGER-EQUATION - SOLITON BEHAVIOR AND BLOWUP

We use Dirac's time-dependent variational principle to discuss several features of the general nonlinear Schrodinger equation i(partial deri vative psi/partial derivative t) + del2psi + \psipsi\(kappa)psi = 0 i n d spatial dimensions for arbitrary nonlinearity parameter kappa. We employ a family of trial variational wave functions, more general than Gaussians, which can be treated analytically and which preserve the c anonical structure (and hence the conservation laws) of the exact syst em. As examples, we derive an approximation to the one-dimensional sol iton solution and demonstrate the ''universality'' of the critical exp onent for blowup in the supercritical case, kappad > 2. For the critic al case kappad = 2, we find that one gets an excellent estimate for th e critical mass necessary for blowup when we minimize the blowup mass with respect to the non-Gaussian variational parameter.