We present a dimension reduction and a numerical method for an effective one dimensional Gross Pitaevskii equation (GPE) for strongly elongated (“cigar shaped”) Bose Einstein condensates. The dimensional reduction from the 3D setting is achieved via a generalized separation ansatz for the wave function into a longitudinal and radial component which is chosen as the groundstate of a radial Schrödinger equation in an anisotropic oscillator confinement. We allow the radial components to implicitly depend on the longitudinal displacement to have an additional degree of freedom in the longitudinal direction and obtain an effective description in 1D by integration over the radial components in the three dimensional action. The Euler Lagrange equations lead to a coupled system of a nonlinear Schrödinger equation (NLS) and a differential algebraic constraint.
This approach was introduced by physicists in a heuristic way, called the Non-Polynomial Schrödinger Equation (NPSE) method, where some terms are neglected that we keep in our more mathematical approach. As a numerical method we chose a time split spectral scheme for the NLS coupled to a finite difference scheme. Finally we present simulations of different system configurations and compare our results to the ones obtained by the NPSE method by reference to a 3D wave function to work out the benefits of our approach.