  Synchronous collisions of solitons of the Korteweg -- de Vries equation are
considered as a representative example of the interaction of a large number of
solitons in a soliton gas. Statistical properties of the soliton field are
examined for a model distribution of soliton amplitudes according to a power
law. $N$-soliton solutions ($N \le 50$) are constructed with the help of a
numerical procedure using the Darboux transformation and 100-digits arithmetic.
It is shown that there exist qualitatively different patterns of evolving
multisoliton solutions depending on the amplitude distribution. Collisions of a
large number of solitons lead to the decrease of values of statistical moments
(the orders from 3 to 7 have been considered). The statistical moments are
shown to exhibit long intervals of quasi-stationary behavior in the case of a
sufficiently large number of interacting solitons with close amplitudes. These
intervals can be characterized by the maximum value of the soliton gas density
and by ``smoothing'' of the wave fields in integral sense. The analytical
estimates describing these degenerate states of interacting solitons are
obtained.
