Nonlinear phenomena accompanying the development of oscillations excited in a layer of a linear dissipative medium by finite displacements of its boundary

A method of describing oscillations in resonators on the basis of evolution equations is proposed. The latter are obtained by simplifying the functional equations under the assumption that the distortions of travelling waves within the resonator length are small, that the Mach number for the moving boundary oscillations is small, and that the frequency is close to one of the natural frequencies of the resonator. The problems of nonstationary oscillations of a layer with a moving boundary are solved. The law that should govern the wall oscillations to provide the development of steady-state linear resonance oscillations is determined. The shape of the resonance curve formed in the presence of a boundary nonlinearity is calculated. The method of matching of asymptotics is applied to the singularly perturbed problem with small dissipation. It is shown that a boundary nonlinearity leads to a distortion of the temporal profile of the standing wave and to the generation of higher harmonics in the process of the development of steady-state oscillations. In contrast to the classical linear problems where the resonance occurs at the coincidence of the external force frequency with one of the natural frequencies, in the case under study the resonance behavior is observed in frequency bands, which are wider the higher the amplitude of the boundary oscillations is.     

Periodic intermittent regime of a boundary flow

Melting of an ultrathin lubricant film during friction between two atomically smooth surfaces is investigated using the Lorentz model for approximating the viscoelastic medium. Second-order differential equations describing damped harmonic oscillations are derived for three boundary relations between the shear stresses, strain, and temperature relaxation times. In all cases, phase portraits and time dependences of stresses are constructed. It is found that under the action of a random force (additive uncorrelated noise), an undamped oscillation mode corresponding to a periodic intermittent regime sets in, which conforms to a periodic stick-slip regime of friction that is mainly responsible for fracture of rubbing parts. The conditions in which the periodic intermittent regime is manifested most clearly are determined, as well as parameters for which this regime does not set in the entire range of the friction surface temperature.     

On the steady-state structure of shock waves in elastic media and dielectrics

A simplified system of equations describing small-amplitude nonlinear quasi-transverse waves in an elastic weakly anisotropic medium with complicated dissipation and dispersion is considered. A simplified system of equations derived for describing the propagation and evolution of one-dimensional weakly nonlinear electromagnetic waves in a weakly anisotropic dielectric is found to be of the same type as the system of equations for quasi-transverse waves in an elastic medium. The steady-state structure of small-amplitude quasi-transverse discontinuities and a large number of admissible discontinuity types is studied using this system of equations. Viscous dissipation is traditionally assumed to be described in terms of the next differentiation order as compared to those constituting the hyperbolic system describing long waves, while the terms responsible for dispersion have an even higher differentiation order.     

Finite-amplitude standing acoustic waves in a cubically nonlinear medium

The acoustic field in a resonator filled with a cubically nonlinear medium is investigated. The field is represented as a linear superposition of two strongly distorted counterpropagating waves. Unlike the case of a quadratically nonlinear medium, the counterpropagating waves in a cubically nonlinear medium are coupled through their mean (over a period) intensities. Free and forced standing waves are considered. Profiles of discontinuous oscillations containing compression and expansion shock fronts are constructed. Resonance curves, which represent the dependences of the mean field intensity on the difference between the boundary oscillation frequency and the frequency of one of the resonator modes, are calculated. The structure of the profiles of strongly distorted “forced” waves is analyzed. It is shown that discontinuities are formed only when the difference between the mean intensity and the detuning takes certain negative values. The discontinuities correspond to the jumps between different solutions to a nonlinear integro-differential equation, which, in the case of small dissipation, degenerates into a third-degree algebraic equation with an undetermined coefficient. The dependence of the intensity of discontinuous standing waves on the frequency of oscillations of the resonator boundary is determined. A nonlinear saturation is revealed: at a very large amplitude of the resonator wall oscillations, the field intensity in the resonator ceases depending on the amplitude and cannot exceed a certain limiting value, which is determined by the nonlinear attenuation at the shock fronts. This intensity maximum is reached when the frequency smoothly increases above the linear resonance. A hysteresis arises, and a bistability takes place, as in the case of a concentrated system at a nonlinear resonance.     

Subwavelength spatial solitons in inhomogeneous Kerr media

Propagation of a narrow beam in a periodic Kerr medium is analyzed for an arbitrary relation between the beam width, modulation period, and optical wavelength. Maxwell’s equations used to obtain solutions describing spatial optical solitons, and their basic properties are determined. The results are compared with those obtained by solving the paraxial nonlinear Schödinger equation. Conditions are found under which these models are mutually inconsistent.     

Boundary layer theory modified for analyzing finite-amplitude oscillations of a viscous liquid charged drop

The existing concepts of the boundary layer arising near the free surface of a viscous liquid, which is related to its periodic motion, are revised with the aim to calculate finite-amplitude linear oscillations of a viscous liquid charged drop. Equations complementing the boundary layer theory are derived for the vicinity of the oscillating free spherical surface of the drop. An analytical solution to these equations is found, comparison with an exact solution is made, and an estimate of the boundary layer thickness is obtained. The domain of applicability of the modified theory is defined.     

Nonlinear processes in media with an acoustic hysteresis and the problems of dynamic interaction between piles and earth foundation

Evolution of a pulsed disturbance in a nonlinear medium whose properties irreversibly vary in the course of wave propagation is studied. Equations describing the propagation process are obtained. It is demonstrated that the waveform distortion and the dynamics of the field and energy characteristics of a signal noticeably differ from those observed in conventional nonlinear media. New nonlinear equations describing a pulse in a medium with relaxation of its nonlinear properties are derived. A finite “delay time” for irreversible processes is introduced in the defining equation. The shape of a pulse reflected from the boundary between an ordinary medium and a nonlinear hereditary medium is calculated. It is demonstrated that, in the case of a fixed relation between the peak pressure in the incident pulse and the ratio of linear impedances of the two media, a total transmission of the trailing edge of the pulse into the compressed medium occurs. Possible applications of the results to topical construction problems are discussed.     

Resonant properties of a nonlinear dissipative layer excited by a vibrating boundary: Q-factor and frequency response

Simplified nonlinear evolution equations describing non-steady-state forced vibrations in an acoustic resonator having one closed end and the other end periodically oscillating are derived. An approach based on a nonlinear functional equation is used. The nonlinear Q-factor and the nonlinear frequency response of the resonator are calculated for steady-state oscillations of both inviscid and dissipative media. The general expression for the mean intensity of the acoustic wave in terms of the characteristic value of a Mathieu function is derived. The process of development of a standing wave is described analytically on the base of exact nonlinear solutions for different laws of periodic motion of the wall. For harmonic excitation the wave profiles are described by Mathieu functions, and their mean energy characteristics by the corresponding eigenvalues. The sawtooth-shaped motion of the boundary leads to a similar process of evolution of the profile, but the solution has a very simple form. Some possibilities to enhance the Q-factor of a nonlinear system by suppression of nonlinear energy losses are discussed.     

A microscopic theory of nucleation. II

Using the nonequilibrium statistical operator obtained in the preceding paper of the authors [1], equations describing the kinetics of nucleation in a nonequilibrium medium are derived. A Fokker-Planck equation is found for embryo distribution functions in the number of particles, energy, momentum, and c.m. coordinates with additional random forces due to non equilibrium processes in the medium. Hydrodynamic equations are obtained for the medium with account of thermodynamic forces due to discontinuities of thermodynamic parameters at the interphase boundary. The symmetry of cross (interphase) kinetic coefficients is considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 44–52, May, 1978.     

A finite element scheme to study the nonlinear optical response of a finite grating without and with defect

We present a simple numerical scheme based on the finite element method (FEM) using transparent-influx boundary conditions to study the nonlinear optical response of a finite one-dimensional grating with Kerr medium. Restricting first to the linear case, we improve the standard FEM to get a fourth order accurate scheme maintaining a symmetric-tridiagonal structure of the finite element matrix. For the full nonlinear equation, we implement the improved FEM for the linear part and a standard FEM for the nonlinear part. The resulting nonlinear system of equations is solved using a weighted-averaged fixed-point iterative method combined with a continuation method. To illustrate the method, we study a periodic structure without and with defect and show that the method has no problem with large nonlinear effect. The method is also found to be able to show the optical bistability behavior of the ideal and the defect structure as a function of either the frequency or the intensity of the input light. The bistability of the ideal periodic structure can be obtained by tuning the frequency to a value close to the bottom or top linear band-edge while that of the defect structure can be produced using a frequency near the defect mode or near the bottom of the linear band-edge. The threshold value can be reduced by increasing the number of layer periods. We found that the threshold needed for the defect structure is much lower then that for a strictly periodic structure of the same length.     

Effect of the spectral profile of a deriving field on the bistability of a dissipative medium

 Through the differential equation describing the behavior of the nonlinear polarization of a medium with respect to an incident field, the Maxwell field equations and the boundary conditions of the field inside a resonator, the relationship between the output and the incident fields is obtained, describing a bistability phenomena. The effect of the spectral profile of the incident field and its spectral halfwidth on the bistability phenomena is studied.     

Modeling of heat explosion with convection

The work is devoted to numerical simulations of the interaction of heat explosion with natural convection. The model consists of the heat equation with a nonlinear source term describing heat production due to an exothermic chemical reaction coupled with the Navier-Stokes equations under the Boussinesq approximation. We show how complex regimes appear through successive bifurcations leading from a stable stationary temperature distribution without convection to a stationary symmetric convective solution, stationary asymmetric convection, periodic in time oscillations, and finally aperiodic oscillations. A simplified model problem is suggested. It describes the main features of solutions of the complete problem.     

Four-wave mixing of quasi-monochromatic waves and few-cycle pulses

Four-wave coherent mixing models for two quasi-monochromatic pumping fields and pulses of a two-component Stokes field with an elliptic polarization and a duration on the order of the period of oscillations have been derived for a two-level medium with a forbidden dipole transition. It is shown that, under the unidirectional wave propagation conditions and in the absence of depletion of pumping, the system of Maxwell-Bloch equations can be reduced to a new completely integrable system of equations. Nonsoliton radiation dynamics of generation of Stokes field pulses is studied in the framework of the integrable reduction of this model. The apparatus for the inverse problem algorithm corresponding to the solvable problem is developed. An approximate asymptotic expression for the leading front of the pulse packet being generated is obtained for various initial and boundary conditions. The application of these results for describing parametric processes involving various types of waves is discussed.     

Flow-induced oscillations of a cantilevered pipe conveying fluid with base excitation

It is known that a plain cantilevered pipe conveying fluid loses its stability by a Hopf bifurcation, leading to either planar or non-planar flutter for flow velocities beyond the critical flow velocity for Hopf bifurcation. If an external mass is attached to the end of the pipe (an end-mass), the resulting dynamics become much richer, showing 2D and 3D quasiperiodic and chaotic oscillations at high flow velocities. In this paper, a cantilevered pipe, with and without an end-mass, subjected to a small-amplitude periodic base excitation is considered. A set of three-dimensional nonlinear equations is used to analyze the pipe?s response at various flow velocities and with different amplitudes and frequencies of base excitation. The nonlinear equations are discretized using the Galerkin technique and the resulting set of equations is solved using Houbolt?s finite difference method. It is shown that for a plain pipe (with no end-mass), non-planar post-instability oscillations can be reduced to planar periodic oscillations for a range of base excitation frequencies and amplitudes. For a pipe with an end-mass, similarly to a plain pipe, three-dimensional period oscillations can be reduced to planar ones. At flow velocities beyond the critical flow velocity for torus instability, the three-dimensional quasiperiodic oscillations can be reduced to two-dimensional quasiperiodic or periodic oscillations, depending on the frequency of base excitation. In all these cases, a low-amplitude base excitation results in reducing the three-dimensional oscillations of the pipe to purely two-dimensional oscillations, over a range of excitation frequencies. These numerical results are in agreement with the previous experimental work.     

Bi-periodic behaviour in the diffusive autocatalator

We consider a reaction-diffusion system, consisting of two coupled PDEs with equal diffusion coefficients, describing autocatalysis under isothermal conditions in a slab. With enforced symmetric boundary conditions only stationary or simple periodic behaviour occurs. When the symmetry is broken by enforcing non-symmetric boundary conditions we have obtained bi-periodic solutions. We have also obtained oscillations essentially confined to one half of the spatial domain.     

On the interaction between a charged drop and an external acoustic field

An interaction between capillary oscillations of a charged drop and an external acoustic field is investigated under conditions in which nonlinear components of the acoustic pressure on the drop surface may be neglected. It is shown that equations describing the temporal evolution of modes of the capillary waves in this case may be either the Mathieu-Hill equations or ordinary inhomogeneous equations of the second order describing forced oscillations. In both cases, the drop instability (of a parametric or resonance type) may result in its disintegration due to deformation caused by the acoustic field at its own drop charge, subcritical in the sense of the Rayleigh criterion.     

Subharmonic generation regimes and chaos fine structure in the classical model of stimulated Raman scattering

Numerical investigation of the system of nonlinear differential second order equations for electron and nuclear oscillations describing stimulated Raman scattering in the model approach has permitted us to discover principally new subharmonic generation regimes and the chaotic response to a regular influence under a certain correlation between the medium and the radiation parameters.     

Anharmonic interactions of elastic and orientational waves in one-dimensional crystals

Planar oscillations of a chain of dumbbell-shaped particles possessing three degrees of freedom are studied. This system models the dynamics of quasi-one-dimensional crystals consisting of elongated anisotropic molecules. A system of nonlinear differential equations describing the anharmonic interaction of the elastic and orientational waves in the lattice, corresponding to different degrees of freedom of the particles, is constructed assuming a cubic interparticle interaction potential. It is shown that in the low-frequency approximation the system obtained is equivalent to the equations of the moment theory of elasticity, widely employed for describing nonlinear and dispersion properties of layered crystals and phase transformations in alloys. Some types of three-wave collinear interactions are investigated, suggesting the possibility of exciting orientational waves in organic crystals because of their nonlinear interaction with acoustic waves. Fiz. Tverd. Tela (St. Petersburg) 39, 137–144 (January 1997)     

The Spatial-Temporal Lamellar Structures in the Confined Ideal Polymer Blends

We consider the formation of self-organized spatial-temporal oscillating structures in symmetric binary polymer blends confined by two flat walls. An influence of these walls on the formation of the oscillating volume structures is studied. This phenomenon is simulated by an initial boundary-value problem for the conserved order parameter (or the concentration of one of the components in a binary mixture). Under a special choice the dynamical Puri-Binder’s boundary conditions these structures look like the lamellar structures. The behavior of the order parameter is described by the modified Cahn-Hilliard equation which models so-called the non-Fickian diffusion in the symmetric binary polymer blends. The nonlinear dynamical boundary conditions correspond to the process of adsorption-desorption on the walls. As a result, these nonlinear surface processes induce into the volume the spatial-temporal asymptotically periodic structures of relaxation, pre-turbulent or turbulent type with finite, countable or non-countable points of discontinuities on the period correspondingly. The frequency of oscillations on the period follows a power-law for the relaxation type and increases exponentially in the other cases.     

Wave instability of a molten metal layer formed by intense laser irradiation

Wave structure of a molten metal layer flowing over the walls of a vapor-gas cavern that appears as an intense laser radiation penetrates deep into condensed media is studied theoretically taking into account surface tension, gravitation, thermocapillary effect, and nonuniform evaporation from the free surface of the melt. A long-wavelength evolution equation describing the evolution of nonlinear waves on the free surface of a plane molten layer is derived. The spatially periodic running solution to this equation is obtained, and the main characteristics (amplitude and period) of the nonlinear wave structures are determined.