  We study the nonlinear mean-field dynamics of molecule formation at coherent
photo- and magneto-association of an atomic Bose-Einstein condensate for the
case when the external field configuration is defined by the quasi-linear level
crossing Demkov-Kunike model, characterized by a bell-shaped pulse and finite
variation of the detuning. We present a general approach to construct an
approximation describing the temporal dynamics of the molecule formation in the
weak interaction regime and apply the developed method to the nonlinear
Demkov-Kunike problem. The presented approximation, written as a scaled
solution to the linear problem associated to the nonlinear one we treat,
contains fitting parameters which are determined through a variational
procedure. Assuming that the parameters involved in the solution of the linear
problem are not modified, we suggest an analytical expression for the scaling
parameter.
