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We describe standing wave patterns induced by an attractive finite-rangedexternal potential inside a large Bose-Einstein Condensate (BEC). As thepotential depth increases, the time independent Gross-Pitaevskii equationdevelops pairs of solutions that have nodes in their wavefunction. We elucidatethe nature of these states and study their dynamical stability. Although westudy the problem in a two-dimensional BEC subject to a cylindrically symmetricsquare-well potential of a radius that is comparable to the coherence length ofthe BEC, our analysis reveals general trends, valid in two and threedimensions, independent of the symmetry of the localized potential well, andsuggestive of the behavior in general, short- and large-range potentials. Oneset of nodal BEC wavefunctions resembles the single particle n node bound statewavefunction of the potential well, the other wavefunctions resemble the n-1node bound-state wavefunction with a kink state pinned by the potential. Thesecond state, though corresponding to the lower free energy value of the pairof n node BEC states, is always unstable, whereas the first can be dynamicallystable in intervals of the potential well depth, implying that the standingwave BEC can evolve from a dynamically unstable to stable, and back to unstablestatus as the potential well is adiabatically deepened, a phenomenon that werefer to as "reentrant dynamical stability".

Reentrant stability of BEC standing wave patterns