We emulate the cubic term ${\Psi }^3$ in the nonlinear Schrödinger equation by a piecewise linear term, thus reducing the problem to a set of uncoupled linear inhomogeneous differential equations. The resulting analytic expressions constitute an excellent approximation to the exact solutions, as is explicitly shown in the case of the kink, the vortex, and a $\delta$ function trap. Such a piecewise linear emulation can be used for any differential equation where the only nonlinearity is a ${\Psi }^3$ one. In particular, it can be used for the nonlinear Schrödinger equation in the presence of harmonic traps, giving analytic Bose-Einstein condensate solutions that reproduce very accurately the numerically calculated ones in one, two, and three dimensions.

• Received 29 January 2003

DOI:https://doi.org/10.1103/PhysRevE.67.066701