Relaxation algorithm to hyperbolic states in Gross-Pitaevskii equation

A new version of the relaxation algorithm is proposed in order to obtain the stationary ground-state solutions of nonlinear Schrödinger-type equations, including the hyperbolic solutions. In a first example, the method is applied to the three-dimensional Gross-Pitaevskii equation, describing a condensed atomic system with attractive two-body interaction in a non-symmetrical trap, to obtain results for the unstable branch. Next, the approach is also shown to be very reliable and easy to be implemented in a non-symmetrical case that we have bifurcation, with nonlinear cubic and quintic terms.