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 derivative psi/partial derivative t) + del2psi + \psi*psi\(kappa)psi = 0 in 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 canonical structure (and hence the conservation laws) of the exact system. As examples, we derive an approximation to the one-dimensional soliton solution and demonstrate the ''universality'' of the critical exponent for blowup in the supercritical case, kappad > 2. For the critical case kappad = 2, we find that one gets an excellent estimate for the critical mass necessary for blowup when we minimize the blowup mass with respect to the non-Gaussian variational parameter.