  We study the dynamics of cold molecule formation via photo- or
magneto-association of quantum degenerate atomic gases for the case when the
field configuration is defined by the quasi-linear level crossing Demkov-Kunike
model, which is characterized by a bell-shaped pulse and finite variation of
the frequency detuning. We generalize the approach developed for the
Landau-Zener model and propose a cubic polynomial equation for the asymptotic
transition probability at infinity applicable in the large detuning regime of
the interaction process. The proposed approximation applies to all values of
the Rabi frequency; hence, it presents a unified description of previously
noticed weak, moderate, and strong interaction limits of the fast sweep regime.
This cubic equation for the transition probability provides improvement of
previous results by an order of the magnitude.
