  We analyze the matter wave transmission above a step potential within the
framework of the cubic-nonlinear Schr\"odinger equation. We present a
comprehensive analysis of the corresponding stationary problem based on an
exact second-order nonlinear differential equation for the probability density.
The exact solution of the problem in terms of the Jacobi elliptic sn-function
is presented and analyzed. Qualitatively distinct types of wave propagation
picture are classified depending on the input parameters of the system.
Analyzing the 2D space of involved dimensionless parameters, the nonlinearity
and the reflecting potential's height/depth given in the units of the chemical
potential, we show that the region of the parameters that does not sustain
restricted solutions is given by a closed curve consisting of a segment of an
elliptic curve and two line intervals. We show that there exists a specific
singular point, belonging to the elliptic curve, which causes a jump from one
evolution scenario to another one. The position of this point is determined and
the peculiarities of the evolution scenarios (oscillatory, non-oscillatory and
diverging) for all the allowed regions of involved parameters are described and
analyzed in detail.
