  Synchronous collisions between a large number of solitons are considered in
the context of a statistical description. It is shown that during the
interaction of solitons of the same signs the wave field is effectively
smoothed out. When the number of solitons increases and the sequence of their
amplitudes decay slower, the focused wave becomes even smoother and the
statistical moments get frozen for a long time. This quasi-stationary state is
characterized by greatly reduced statistical moments and by the density of
solitons close to some critical value. This state may be treated as the
small-dispersion limit, what makes it possible to analytically estimate all
high-order statistical moments. While the focus of the study is made on the
Korteweg--de Vries equation and its modified version, a much broader
applicability of the results to equations that support soliton-type solutions
is discussed.
