Probabilistic characteristics of nonlinear waves in nondispersive media of the hydrodynamic type.

Abstract

The paper considers the probability distributions of the nonlinear wave characteristics in nondispersive media that satisfy the Riemann- and the Kardar-Parisi-Zhang-type equations. By using the Lagrangian and Euler relations of statistical descriptions, expressions are obtained for the probability density of the Riemann wave (displacement) integral through the initial probability density of displacement, velocity, and acceleration. The case of Gaussian initial statistics is considered when multivalued sections in nonlinear waves arise at arbitrarily small distances from the entrance. The expressions obtained in this case should be interpreted as the relative residence time of the process in a certain displacement range. It is shown that, due to the Riemann equation locality, the appearance of ambiguity in the wave profile, which occurs mainly at negative values, does not affect the probability density form at positive bias values.