We consider systems governed by the Gross Pitaevskii equation (GPE) with the Morse potential
V(x)=D(e−2ax−2e−ax) as the trapping potential. For positive values of the coupling constant
g of the cubic term in GPE, we find that the critical value
gc beyond which there are no bound states scales as
D3/4 (for large
D). Studying the quantum evolution of wave packets, we observe that for
ggc, on the otherhand, all initial wave packets escape from the potential and the dynamics is like that of a quantum free particle. For
g<0, we find that there can be initial conditions for which the escaping wave packet can propagate with very little change in width i,e., it remains almost shape invariant.