Development of stochastic oscillations in a one-dimensional dynamical system described by the Korteweg de Vries equation

The behavior of the solution of the Korteweg-de Vries equation for large-sc ale oscillating aperiodic initial conditions prescribed on the entire x axi s is considered. It is shown that the structure of small-scale oscillations arising in a Korteweg-de Vries system as t --> infinity loses its dynamica l properties as a consequence of phase mixing. This process can be called t he generation of soliton turbulence. The infinite system of interacting sol itons with random phases developing under these conditions leads to oscilla tions having a stochastic character. Such a system can be described using t he terms applied to a continuous random process, the probability density an d correlation function. It is shown that for this it suffices to determine from the prescribed initial conditions amplitude distribution function of t he solitons and their mean spatial density. The limiting stochastic charact eristics of the mixed state for problems with initial data in the form of a n infinite sequence of isolated small-scale pulses are found. Also, the pro blem of stochastic mixing under arbitrary initial conditions in the dispers ionless limit (the Hopf equation) is completely solved. (C) 1999 American I nstitute of Physics. [S1063-7761(99)02701-8].