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We consider systems governed by the Gross Pitaevskii equation (GPE) with theMorse potential $V(x)=D(e^{-2ax}-2e^{-ax})$ as the trapping potential. Forpositive values of the coupling constant $g$ of the cubic term in GPE, we findthat 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, weobserve that for $g<g_c$, the initial wave packet needs a critical momentum forthe packet to escape from the potential. For $g>g_c$, on the otherhand, allinitial wave packets escape from the potential and the dynamics is like that ofa quantum free particle. For $g<0$, we find that there can be initialconditions for which the escaping wave packet can propagate with very littlechange in width i,e., it remains almost shape invariant.

Gross Pitaevskii Equation with a Morse potential: bound states and evolution of wave packet