  We study nonlinear mean-field dynamics of ultracold molecule formation in the
case when the external field configuration is defined by the level-crossing
Demkov-Kunike model, characterized by a bell-shaped coupling and finite
variation of the detuning. Analyzing the fast sweep rate regime of the strong
interaction limit, which models a situation when the peak value of the coupling
is large enough and the resonance crossing is sufficiently fast, we construct a
highly accurate ansatz to describe the temporal dynamics of the molecule
formation in the mentioned interaction regime. The absolute error of the
constructed approximation is less than 3*10^-6 for the final transition
probability while at certain time points it might increase up to 10^-3.
Examining the role of the different terms in the constructed approximation, we
prove that in the fast sweep rate regime of the strong interaction limit the
temporal dynamics of the atom-molecule conversion effectively consists of the
process of resonance crossing, which is governed by a nonlinear equation,
followed by atom-molecular coherent oscillations which are basically described
by a solution of the linear problem, associated with the considered nonlinear
one.
