Development of stochastic oscillations in a one-dimensional dynamical system described by the Korteweg-de Vries equation |
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Authors: | A V Gurevich K P Zybkin G A Él’ |
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Institution: | (1) I. E. Tamm Division of Theoretical Physics, P. N. Lebedev Physical Institute, Russian Academy of Science, 117924 Moscow, Russia;(2) Institute of Earth Magnetism, the Ionosphere, and Radio Wave Propagation, Russian Academy of Sciences, 142092 Troitsk, Moscow Oblast’, Russia |
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Abstract: | The behavior of the solution of the Korteweg-de Vries equation for large-scale oscillating periodic initial conditions prescribed
on the entire x axis is considered. It is shown that the structure of small-scale oscillations arising in a Korteweg-de Vries system as t→∞ loses its dynamical properties as a consequence of phase mixing. This process can be called the generation of soliton turbulence.
The infinite system of interacting solitons with random phases developing under these conditions leads to oscillations having
a stochastic character. Such a system can be described using the terms applied to a continuous random process, the probability
density and correlation function. It is shown that for this it suffices to determine from the prescribed initial conditions
amplitude distribution function of the solitons and their mean spatial density. The limiting stochastic characteristics of
the mixed state for problems with initial data in the form of an infinite sequence of isolated small-scale pulses are found.
Also, the problem of stochastic mixing under arbitrary initial conditions in the dispersionless limit (the Hopf equation)
is completely solved.
Zh. éksp. Teor. Fiz. 115, 333–360 (January 1999) |
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