Probability Distributions of the Riemann Wave and an Integral of It
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Probability Distributions of the Riemann Wave and an Integral of It S. N. Gurbatova,* and E. N. Pelinovskyb,c,d,** Presented by Academician E.A. Kuznetsov April 23, 2020 Received April 25, 2020; revised April 30, 2020; accepted April 30, 2020
Abstract—In this paper, we study the statistical characteristics of the Riemann wave and the integral of it. Such problems arise in the physics of fronts’ propagation, in particular, during the wave run-up on the shore, a flame front, and phase surfaces. Using the relation of the Lagrangian and Eulerian statistical descriptions, we obtained general expressions for the probability distributions of the front velocity and its displacement. We show that the joint probabilistic distribution of displacement, velocity, and acceleration at the input of a nonlinear medium are necessary to find the probability distribution of displacement. Based on the idea of probability as the time the signal spent in a certain interval, the characteristics of the waves after their breaking were calculated. Keywords: nonlinear waves in the nondispersive media, Riemann equation, Kardar–Parisi–Zang equation, probability characteristics of nonlinear waves DOI: 10.1134/S1028335820080029
The dynamics of random Riemann waves is well studied in many media, especially in nonlinear acoustics (see, for example, [1–3]). At the same time, in a number of problems, it has become necessary to study the random characteristics of the integrals of Riemann waves. In particular, in the problem of the rolling of sea waves to the coast, the Riemannian solution describes the velocity of the leading edge of the water shaft, and the integral of it is the displacement of the water edge, which defines the boundaries of the flood zone [4–7]. In problems of describing expanding surfaces (phase fronts in geometric optics, flame front, etc.), the propagation velocity of fronts is also described by the Riemann wave, and the phase is described by the integral of it. The Kardar–Parisi– Zhang equation [8–10] is a popular model here. In cosmology, the Zel’dovich approximation, which describes the initial nonlinear stage of gravitational
instability, is widely applied when describing the largescale structure of the Universe. In this case, the particle motion in the corresponding variables is reduced to the Riemann equation, while the evolution of the velocity and potential fields is equivalent to the evolution of the optical wave behind the phase screen [11, 12]. In fact, the probability distributions of the integrals of Riemann waves have not yet been studied. Here, we obtain formulas describing the probability of the distribution of the integral of the Riemann wave, including the initial stage of wave collapse (the so-called gradient catastrophe). BASIC EQUATIONS The equations describing the Riemann waves and the integrals of them can be written as follows:
∂u − u ∂u = 0, ∂x ∂t
a Lobachevsky
State University of Nizhny Novgorod, Nizhny Novgorod, 603105 Russia b Nizhny Novgorod State Technical University n.a. R. Alekseev, Nizhny
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