  We present a method, the dynamical cumulant expansion, that allows to
calculate quantum corrections for time-dependent quantities of interacting spin
systems or single spins with anisotropy. This method is applied to the
quantum-spin model \hat{H} = -H_z(t)S_z + V(\bf{S}) with H_z(\pm\infty) =
\pm\infty and \Psi (-\infty)=|-S> we study the quantity P(t)=(1-<S_z>_t/S)/2.
The case V(\bf{S})=-H_x S_x corresponds to the standard
Landau-Zener-Stueckelberg model of tunneling at avoided-level crossing for N=2S
independent particles mapped onto a single-spin-S problem, P(t) being the
staying probability. Here the solution does not depend on S and follows, e.g.,
from the classical Landau-Lifshitz equation. A term -DS_z^2 accounts for
particles' interaction and it makes the model nonlinear and essentially quantum
mechanical. The 1/S corrections obtained with our method are in a good accord
with a full quantum-mechanical solution if the classical motion is regular, as
for D>0.
