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Higher order transmission resonances in above-barrier reflection of ultra-cold atoms

The reflectionless transmission resonances in above-barrier reflection of Bose-Einstein condensates by the Rosen-Morse potential are considered using the mean field Gross-Pitaevskii approach. Applying an exact third order nonlinear differential equation obeyed by the condensate's density, the exact solution of the problem for the first resonance is derived. It is shown that in the nonlinear case the total transmission is possible for positive potential heights, i.e., for potential barriers. Further, it is shown that an appropriate approximation for higher-order resonances can be constructed using a limit solution of the equation for the density written as a root of a polynomial equation of the third degree. Using this limit function and the solution for the first resonance, a simple approximation for the shift of the nonlinear resonance potential's depth from the corresponding linear resonance's position is constructed for higher order resonances. The result is written as a linear function of the resonance order. This behavior notably differs from the case of the rectangular barrier for which the nonlinear shift is approximately constant for all the resonance orders.