Ground state of the one dimensional Gross-Pitaevskii equation with a Morse potential

We have studied the ground state of the Gross-Pitaevskii equation (nonlinear Schrodinger equation) for a Morse potential via a variational approach. It is seen that the ground state ceases to be bound when the coupling constant of the nonlinear term reaches a critical value. The disappearence of the ground state resembles a saddle node bifurcation.