The asymptotic behavior of the solutions of the KdV equation in the classic
al limit with an oscillating nonperiodic initial function u(0)(x) prescribe
d on the entire x axis is investigated. For such an initial condition, nonl
inear oscillations, which become stochastic in the asymptotic limit t --> i
nfinity, develop in the system. The complete system of conservation laws is
formulated in the integral form, and it is demonstrated that this system i
s equivalent to the spectral density of the discrete levels of the initial
problem. The scattering problem is studied for the Schrodinger equation wit
h the initial potential -u(0)(x), and it is shown that the scattering phase
is a uniformly distributed random quantity. A modified method is developed
for solving the inverse scattering problem by constructing the maximizer f
or an N-soliton solution with random initial phases. A one-to-one relation
is established between the spectrum of the discrete levels of the initial s
tate of the system and the spectrum established in phase space. It is shown
that when the system passes into the stochastic state, all KdV integral co
nservation laws are satisfied. The first three laws are satisfied exactly,
while the remaining laws are satisfied in the WKB approximation, i.e., to w
ithin the square of a small dispersion parameter. The concept of a quasisol
iton, playing in the stochastic state of the system the role of a standard
soliton in the dynamical limit, is introduced. A method is developed for de
termining the probability density f(u), which is calculated for a specific
initial problem. Physically, the problem studied describes a developed one-
dimensional turbulent state in dispersion hydrodynamics. (C) 2000 MAIK "Nau
ka/Interperiodica".