Nonlinear modes for the Gross–Pitaevskii equation—a demonstrative computation approach

A method for the study of steady-state nonlinear modes for the Gross–Pitaevskii equation (GPE) is described. It is based on the exact statement about the coding of the steady-state solutions of GPE which vanish as x → +∞ by reals. This allows us to fulfil the demonstrative computation of nonlinear modes of GPE, i.e. the computation which allows us to guarantee that all nonlinear modes within a given range of parameters have been found. The method has been applied to GPE with quadratic and double-well potentials, for both repulsive and attractive nonlinearities. The bifurcation diagrams of nonlinear modes in these cases are represented. The stability of these modes has been discussed.