  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 $g_c$ beyond which there are no bound states scales as
$D^{3/4}$ (for large $D$). Studying the quantum evolution of wave packets, we
observe that for $g<g_c$, the initial wave packet needs a critical momentum for
the packet to escape from the potential. For $g>g_c$, 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.
