Statistical limit in a completely integrable system with deterministic initial conditions

The asymptotic behavior of the solutions of the KdV equation in the classical limit with an oscillating nonperiodic initial function u ₀(x) prescribed on the entire x axis is investigated. For such an initial condition, nonlinear oscillations, which become stochastic in the asymptotic limit t→∞, develop in the system. The complete system of conservation laws is formulated in the integral form, and it is demonstrated that this system is equivalent to the spectral density of the discrete levels of the initial problem. The scattering problem is studied for the Schrödinger equation with 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 for 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 state 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 conservation laws are satisfied. The first three laws are satisfied exactly, while the remaining laws are satisfied in the WKB approximation, i.e., to within the square of a small dispersion parameter. The concept of a quasisoliton, 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 determining 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.