Variational approximations to the nonlinear Schrödinger equation

Abstract

We study analytically the family of nonlinear Schrödinger equations (NLSE) iϖψ ϖ + ϖ 2ψ ϖx 2 +g… ψψ… K ψ = 0 for arbitrary values of the nonlinearity parameter ϰ using two variational strategies based on two different perturbation expansions of the equation - the delta expansion (DE) (which expands in the nonlinearity) and the linear delta expansion (LDE) (which expands about an optimal linear term). For simplicity we discuss solutions to the initial value problem for which ψ( x, t=0) = Cδ( x). For the LDE the variational calculation gives the exact analytic solution to the NLSE for arbitrary ϰ for this initial data. The answer has different structure for ϰ< 1, ϰ=1 and ϰ > 1. For the delta expansion the variational calculation gives a sequence of approximations to the solution valid for ϰ<1.