Stationary Quantum States Associated with Nonlinear Scattering in One Dimension

Quantum scattering dynamics can be reduced to a description in terms of stationary states. In the presence of nonlinearity, this is not typically possible, but stationary states exist and provide a framework for understanding the dynamics. We do a comprehensive analysis of the quantum states for a step potential and for a barrier potential in one dimension in the presence of nonlinearity, such as induced by interatomic interactions for coherent quantum states associated for example with Bose-Einstein condensates (BEC). In the mean field limit, using analytical expressions involving Jacobi elliptic functions, we find the full range of allowed solutions, which span a substantially larger variety than in the linear case used to model scattering in one dimension. We use a system based on the nature of the roots of the hydrodynamic equations to classify and characterize the solutions and determine the associated range of allowed physical parameters.  We consider these stationary solutions in the context of nonlinear scattering dynamics at potential barriers.  The intricate sensitivity of the stationary solutions to small changes in the physical parameters in certain critical regimes open possibilities of high precision metrology based on quantum coherent states.