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

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.

Figure 1

The velocity profile (the upper figure) and the displacement (the lower figure) in the Lagrangian (the dashed line) and the Euler representations (the solid line) for $\xi =1.8$.

Figure 2

The Lagrangian (the upper figure) and the Euler (the lower figure) displacement probability density at $l=0$, (the dashed line), $l=0.5$ (points), and $l=1$ (the solid line).

Figure 3

The Euler displacement probability density (relative residence time) at $l=1$ for two models when ambiguities are taken into account [(a) $\left|q\right|=0.9$, (b) $\left|q\right|=0.1$]. The solid line shows the displacement probability distribution when the intervals after breaking are taken into account with the minus sign. The dotted line shows the distribution when these intervals are not taken into consideration.

Figure 4

The Euler potential probability density (relative residence time) for $l=1$ and $\left|q\right|=0.9$ for two models when ambiguities are taken into account. The solid line shows the probability distribution ((4.16)), when the intervals after breaking are taken into account with the minus sign. The dashed line shows the distribution ((4.25)) when all the intervals in the polysemy region are taken into account with the positive sign.