The nonlinear dynamics of a crystal lattice where the atoms are positioned along parallel rods is studied. They may move only in one direction and this constraint leads to the appearance of nonlinearity even the forces between the atoms obey the linear Hooke's law. This nonlinearity turns out to be strong. The equations of motion of the individual lattice atoms are written, and. in the continuum limit when the lattice period is small in comparison with the wavelength, a new strongly nonlinear partial differential equation is derived. The waves traveling in the direction orthogonal to the rods are purely transverse slow waves, governed by an equation of the Heisenberg type. In the direction along the rods, a fast purely longitudinal wave can propagate. In general, when the wave travels at an arbitrary angle, it is neither purely longitudinal nor transverse and the periodic structure exhibits anisotropic properties. Their velocity depends strongly on the direction of propagation and the structure exhibits properties similar to a skeletal muscle with stretched fibers. Special attention is paid to the soliton solutions of this equation and their behavior is studied. For non-stationary quasi-longitudinal waves, a new evolution equation, rich in symmetries, is derived. One of the solutions with a fixed transverse structure is described by elliptic integrals and evolves in accordance with a cubic nonlinear equation of the Klein–Gordon type. © 2019 Elsevier B.V.

We have studied the dynamics of an artificial nonlinear element representing a flexible membrane with oscillation limiters and a static pressing force. Such an element has the property of “bimodularity” and demonstrates “modular” nonlinearity. We have constructed a mathematical model that describes these oscillations. Their shapes have been calculated. We follow the analogy with a classical object—Galileo’s pendulum. We demonstrate that for a low-frequency excitation of the membrane, the level of the harmonics in the spectrum is higher than in the vicinity of the resonance frequency. We have established a strong dependence of the level of the harmonics on the magnitude of the pressing force for a weak perturbation. We propose a design scheme for a device in the quasi-static approximation possessing the property of bimodularity. We perform an experiment that confirms its operability. We show a qualitative coincidence of the experimental results and calculations when detecting an amplitude-modulated signal. © 2018, Pleiades Publishing, Ltd.

The observed nonclassical power-law dependence of the amplitude of the second harmonic wave on the amplitude of a harmonic pump wave is explained as a phenomenon associated with two types of nonlinearity in a structurally inhomogeneous medium. An approach to solving the inverse problem of determining the nonlinearity parameters and the exponent in the above-mentioned dependence is demonstrated. To describe the effects of strongly pronounced nonlinearity, equations containing a double nonlinearity and generalizing the Hopf and Burgers equations are proposed. The possibility of their exact linearization is demonstrated. The profiles, spectral composition, and average wave intensity in such doubly nonlinear media are calculated. The shape of the shock front is found, and its width is estimated. The wave energy losses that depend on both nonlinearity parameters—quadratic and modular—are calculated. © 2018, Pleiades Publishing, Ltd.

The interaction of weak noise and regular signals with a shock wave having a finite width is studied in the framework of the Burgers equation model. The temporal realization of the random process located behind the front approaches it at supersonic speed. In the process of moving to the front, the intensity of noise decreases and the correlation time increases. In the central region of the shock front, noise reveals non-trivial behaviour. For large acoustic Reynolds numbers the average intensity can increase and reach a maximum value at a definite distance. The behaviour of statistical characteristics is studied using linearized Burgers equation with variable coefficients reducible to an autonomous equation. This model allows one to take into account not only the finite width of the front, but the attenuation and diverse character of initial profiles and spectra as well. Analytical solutions of this equation are derived. Interaction of regular signals of complex shape with the front is studied by numerical methods. Some illustrative examples of ongoing processes are given. Among possible applications, the controlling the spectra of signals, in particular, noise suppression by irradiating it with shocks or sawtooth waves can be mentioned. © 2018 Elsevier B.V.

Topic and aim. A brief review of publications and discussion of some mathematical models are presented, which, in the author's opinion, are well-known only to a few specialists. These models are not well studied, despite their universality and practical significance. Since the results were published at different times and in different journals, it is useful to summarize them in one article. The goal is to form a general idea of the subject for the readers and to interest them with mathematical, physical or applied details described in the cited references. Investigated models. Higher-order dissipative models are discussed. Precisely linearizable equations containing nonanalytic nonlinearities -quadratically-cubic (QC) and modular (M) -are considered. Equations like Burgers, KdV, KZ, Ostrovsky-Vakhnenko, inhomogeneous and nonlinear integro-differential equations are analyzed. Results. The appearance of dissipative oscillations near the shock front is explained. The formation in the QC-medium of compression and rarefaction shocks, which are stable only for certain parameters of the «jump», as well as the formation of periodic trapezoidal sawtooth waves and self-similar N-pulse signals are described. Collisions of single pulses in the M-medium are discussed, revealing new corpuscular properties (mutual absorption and annihilation). Collisions are similar to inter-actions of clusters of chemically reacting substances, for example, fuel and oxidizer. The features of the behavior of «modular» solitons are described. The phenomenon of nonlinear wave resonance in media with QC-, Q-and M-nonlinearities is studied. Precisely linearizable inhomogeneous equations with external sources are used. The shift of maximum of resonance curves relative to the linear position, which is determined by the equality of velocities of freely propagating and forced waves, is indicated. Simplified models for diffracting beams obtained by projecting 3D equations onto the beam axis are analyzed. Strongly nonlinear waves in systems with holonomic constraints are discussed. Integro-differential equations with relaxation type kernel, and the possibility of reducing them to differential and differential-difference equations are considered. Discussion. The material is outlined on a popular level. Apparently, these studies can be continued if the readers find them interesting enough. © 2018 Saratov State University. All rights reserved.

Everybody is accustomed to that nonlinear effects amplify with increasing amplitude or intensity of a wave. When the amplitude becomes small, the nonlinearity disappears and the wave enters a linear regime. Instead, we shall consider here so-called strong nonlinearity of the first type (according to a classification introduced earlier by the authors) where the effects of nonlinearity do not disappear even for infinitesimal amplitudes. Among these nonlinearities are modular (M) and quadratically-cubic (QC). When these nonlinearities are included in partial differential equations, they form new mathematical models describing new physical effects. Such equations have been proposed over the past few years and a review of these models is given here. They are interesting because of two reasons: (i) the equations admit exact analytic solutions, and (ii) the solutions describe real physical phenomena. Among them are M- and QC-equations of Hopf, Burgers, Korteveg-de Vries, Khokhlov-Zabolotskaya and Ostrovsky-Vakhnenko types. Media with non-analytic nonlinearities exist among composites, meta-materials, and inhomogeneous and multiphase systems. Some of the physical phenomena manifested in such media are described, e.g. stable shock fronts of compression and rarefaction in QC-media. The last cannot exist in usual media and the periodic wave consists of a series of trapezoidal teeth, rather than usual triangular. In an M-nonlinear medium collision, mutual losses and annihilation of pulses are studied. These pulses exhibit corpuscular properties and, in contrast to solitons (elastic particles) and shock waves (absolutely inelastic collisions), they behave like clots of chemical reagents (fuel and oxidizer). As result of an reaction, the smaller component disappears, and the larger decreases. At equal "masses", these clots disappear or annihilate. In M-media a new stable wave - a modular soliton - exists. Other interesting physical phenomena occur for focused waves in M-media and a review of these is also included in the presentation. Copyright © (2018) by International Institute of Acoustics & Vibration.All rights reserved.

A singularly perturbed initial-boundary value problem for a parabolic equation known in applications as a Burgers-type or reaction-diffusion-advection equation is considered. An asymptotic approximation of solutions with a moving front is constructed in the case of modular and quadratic nonlinearity and nonlinear amplification. The influence exerted by nonlinear amplification on front propagation and blowing- up is determined. The front localization and the blowing-up time are estimated.

The review of new mathematical models containing non-analytic nonlinearities is given. These equations have been proposed recently, over the past few years. The models describe strongly nonlinear waves of the first type, according to the classification introduced earlier by the authors. These models are interesting because of two reasons: (i) equations admit exact analytic solutions, and (ii) solutions describe the real physical phenomena. Among these models are modular and quadratically cubic equations of Hopf, Burgers, Korteveg-de Vries, Khokhlov-Zabolotskaya and Ostrovsky-Vakhnenko type. Media with non-analytic nonlinearities exist among composites, meta-materials, inhomogeneous and multiphase systems. Some physical phenomena manifested in the propagation of waves in such media are described on the qualitative level of severity.

A one-dimensional equation is presented that generalizes the Burgers equation known in the theory of waves and in turbulence models. It describes the nonlinear evolution of waves in pipes of variable cross section filled with a dissipative medium, as well as in ray tubes, if the approximation of geometric acoustics of an inhomogeneous medium is used. The generalized equation is reduced to the common Burgers equation with a dissipative parameter-the "Reynolds-Goldberg number," depending on the coordinate. The method for solving statistical problems corresponding to specified characteristics of a noise signal at the input of the system is described. Integral expressions for exact solutions are given for the correlation function and the noise intensity spectrum experiencing nonlinear distortions during propagation in a waveguide. For waves in a dissipative medium, an approximate method of calculating statistical characteristics is given, consisting in finding an auxiliary correlation function and the subsequent nonlinear functional transformation. Solutions have a complicated form, so physical analysis of phenomena requires the numerical methods. For some correlation functions of stationary noise with initial Gaussian statistics and some waveguide systems, it is possible to obtain simple results.

We study the interaction of noise and regular signals with a front whose steepness increases or decreases owing to nonlinear distortion of the profile of an intense pumping wave. Projective transformation is used, which is a result of one of the Burgers equation symmetries. Signal interaction with the pumping wave at its leading edge results in an increase in signal amplitude, a decrease in its time scale, an increase in the signal evolution rate, and earlier merging of discontinuities. At the trailing edge, an increase in signal amplitude, an increase in the time scale, and deceleration of the evolution rate occur. Formulas are obtained that describe the transformation of the spectrum and the correlation function of noise. Laws of the change in noise energy for both small and large Reynolds numbers are found. We study the interaction of weak noise with a nonstationary shock front in a medium with a finite viscosity. It is shown that, owing to competition between amplification at the shock front and high-frequency attenuation, the dependence on the noise intensity on distance has a nonmonotonic character, and at large distances, the intensity tends to zero, while the correlation time tends to a finite value.