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| For the stationary Gross--Pitaevskii equation with harmonic real and linear imaginary potentials in the space of one dimension, we construct the ground state solutions in the limit of large densities (large chemical potentials), where the solution degenerates into a compact Thomas--Fermi approximation. We find that the construction of the Thomas--Fermi approximation can be achieved with an invertible coordinate transformation and an unstable manifold theorem for planar dynamical systems. The persistence and justification of the
Thomas-Fermi approximation are achieved with another coordinate transformation and reduction of the underlying problem to the Painlev\'e-II equation with a unique global Hastings-McLeod solution. Extensions of our results are discussed in the context of the stationary Gross--Pitaevskii equation with harmonic real and localized odd imaginary potentials. |
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