We investigate the mean-field scattering dynamics of a quasi-one-dimensional Bose-Einstein condensate interacting with a Rosen-Morse (RM) potential. For specific potential and nonlinearity parameters, we derive exact degenerate reflectionless states (with twofold or threefold degeneracy). Using the Bogoliubov–de Gennes approach, we analyze the stability of these reflectionless degenerate states, demonstrating that only one solution within each degenerate manifold is dynamically stable. Furthermore, we study a configuration with spatially localized nonlinearity, identifying an exact reflectionless state under specific conditions. Using two numerical approaches, we map the transmission input-output response of the model with spatially localized nonlinearity and identify a bistable window with hysteresis. For the parameter set studied, the exact reflectionless state coincides with the upper turning point of the $\mathrm{S}$-shaped response curve. Taken together, our analytical and numerical results show bistable transmission in the RM-based nonlinear-scattering models considered here. Given that RM potentials are experimentally feasible within ultracold atomic systems and engineered photonic lattices, our theoretical predictions could be validated on these platforms.

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• Nonlinear resonance
• Optical bistability