Solution of the Fluid Dynamical Formulation of Nonlinear Schrodinger Equation with Radial Basis Function Interpolation

摘要

Recently, interest in de Broglie-Bohm formulation of quantum mechanics has increased dramatically within various fields. With the wave function written in Madelung's form, the de Broglie-Bohm theory possesses an intuitive physical representation as quantum fluid dynamics (QFD), reminiscent of classical fluid dynamics. Besides its conceptual importance, the potential numerical advantage of the QFD formulation over working with the Schroedinger equation can be exploited to provide an alternative solution method. Radial basis function (RBF) interpolation has been widely used as a spatial approximation scheme for PDEs. The RBF is a very high order scheme, one can obtain excellent results using a coarser distribution of data points than with a finite difference (FD) approximation. It has been demonstrated multiquadrics (MQ) is an excellent spatial approximation scheme for nonlinear system of hyperbolic PDEs. Applications of RBF to QFD based on linear Schroedinger equation have been recently presented. In this work, we consider the RBF scheme for QFD based on Gross-Pitaevskii(GP) equation which is a nonlinear Schroedinger equation (NLSE). After substituting the polar form of the wave function into the GP equation, one obtains a set of governing equations reminiscent of compressible gas dynamics.