SEPARABILITY AND STABILITY OF SOLUTIONS OF THE 1-DIMENSIONAL NLSE WITH RESPECT TO EXTENSION INTO 2 AND 3 DIMENSIONS, AND TO INITIAL PERTURBATION BY WHITE NOISE

Abstract

Stationary states of the nonlinear Schrödinger equation (NLSE) found analytically in previous work are extended into 2 and 3 dimensions by the simplest possible ansatz: namely, it is assumed that the direct product of one dimensional solutions for each dimension will yield a stationary state. The solutions considered mimic the dynamics of a repulsive Bose-Einstein condensate (BEC) in a trap of high aspect ratio. This assumption of separability, as established by direct numerical integration of the NLSE via variable step 4th order Runge-Kutta using a pseudo spectral basis, is found to work well for both ground and excited states for box transverse confinement, and for either box or periodic boundary conditions along the longest trap axis. Addition of white noise at t = 0, followed by similar numerical propagation in either 2 or 3 dimensions, is found to lead to instability once the transverse confining dimension are greater than approximately 6 healing lengths. Such instabilites eventually manifest themselves as vortices fathered by the well known snake instability of the NLSE solitons in dimensionalities higher than 1. The dynamics of interacting solitons may become chaotic as the solitons themselves become unstable in the presence of noise.