Bistability and Exact Reflectionless States in Nonlinear Scattering of a Bose--Einstein Condensate

TL;DR

This paper analytically explores the scattering behavior of a Bose--Einstein condensate in a nonlinear potential, revealing exact reflectionless states, their stability, and conditions for bistability in matter-wave transport.

Contribution

It provides an analytical framework for understanding multistable transmission states and their stability in nonlinear Bose--Einstein condensate scattering.

Findings

Exact degenerate reflectionless states identified

Only one degenerate state is dynamically stable

Transition from monostability to bistability with increasing wave amplitude

Abstract

We investigate the mean-field scattering dynamics of a quasi-one-dimensional Bose--Einstein condensate interacting with a Rosen--Morse potential. For specific potential and nonlinearity parameters, we derive analytically exact, degenerate scattering states (doubly or triply degenerate) exhibiting perfect transmission. 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. Numerical analysis shows that this state marks the system's transition from monostability to bistability as the incident wave amplitude increases. Our work establishes an analytic framework for these multistable transmission phenomena, directly relevant to coherent matter-wave transport in ultracold atomic systems and optical propagation in engineered photonic lattices.