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

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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 g<gc, the initial wave packet needs a critical momentum for the packet to escape from the potential. For g>gc, 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.