  We study the strong coupling limit of a quadratic-nonlinear Landau-Zener
problem for coherent photo- and magneto-association of cold atoms taking into
account the atom-atom, atom-molecule, and molecule-molecule elastic scattering.
Using an exact third-order nonlinear differential equation for the molecular
state probability, we develop a variational approach which enables us to
construct a highly accurate and simple analytic approximation describing the
time dynamics of the coupled atom-molecule system. We show that the
approximation describing time evolution of the molecular state probability can
be written as a sum of two distinct terms; the first one, being a solution to a
limit first-order nonlinear equation, effectively describes the process of the
molecule formation while the second one, being a scaled solution to the linear
Landau-Zener problem (but now with negative effective Landau-Zener parameter as
long as the strong coupling regime is considered), corresponds to the remaining
oscillations which come up when the process of molecule formation is over.
