Nonlinear equations for quasi-monochromatic waves near a caustic in a dispersive dissipative medium with cubic or quadratic nonlinearity

The behavior of a quasi-monochromatic nonlinear wave near a caustic is considered. Nonlinear ordinary differential equations for a dispersive dissipative medium with a cubic or quadratic nonlinearity are derived. For the latter medium, nonstationary equations describing it near the caustic are presented with allowance for the dissipative dispersive terms. These equations yield ordinary ones for quasi-monochromatic waves. The amplitude of the second harmonic is expressed in terms of the squared amplitude of the first harmonic. The amplitude of the second harmonic, as well as the solution as a whole, increases near the caustic.     

Instability and evolution of nonlinearly interacting water waves

We consider the modulational instability of nonlinearly interacting two-dimensional waves in deep water, which are described by a pair of two-dimensional coupled nonlinear Schr?dinger equations. We derive a nonlinear dispersion relation. The latter is numerically analyzed to obtain the regions and the associated growth rates of the modulational instability. Furthermore, we follow the long term evolution of the latter by means of computer simulations of the governing nonlinear equations and demonstrate the formation of localized coherent wave envelopes. Our results should be useful for understanding the formation and nonlinear propagation characteristics of large-amplitude freak waves in deep water.     

Parametric excitation of traveling waves in a circular non-dispersive medium

This paper discusses a special type of propagating waves created by parametric excitation in a circular taught string. The string, being a non-dispersive medium propagates deformations in a similar manner to electromagnetic waves in vacuum, both have simple wavelength–frequency relationship that play an important role here. Nonlinear equations are derived under the assumption of finite deformations, whose solution produces a square-wave like, limited-amplitude, traveling wave. Closed-form expressions are obtained for the parametric excitation characteristics of the nonlinear system and the steady-state traveling waves are described by a generalized eigenvalue problem. The latter relates the nonlinear elongation of the neutral axis to the participating wavelengths forming the propagating wave. Detailed numerical simulations are provided to validate the solution and to illustrate graphically the waveforms. It is shown that propagating sinusoidal parametric excitation gives rise to various square-wave like deformation shapes which a unique phenomenon is arising in non-dispersive media.     

Nonlinear wave interactions in porous media filled with a quantum fluid

The problem of the nonlinear interaction between the fourth sound and an acoustic wave propagating in a porous medium filled with superfluid helium is solved. Based on the Landau equations of quantum fluid dynamics and on the Biot theory of mechanical waves in a porous medium, nonlinear wave equations are derived for studying the aforementioned interaction. An expression is obtained for the vertex that determines the excitation of an acoustic wave by two waves of the fourth sound. The possibility of an experimental observation of this process is estimated.     

A representation of Kerr-like nonlinearity for the analytical TM surface polariton solution

The TM-polarized waves propagating along the interface between a nonlinear Kerr-like material and linear cladding are investigated. We analyse the nonlinear dielectric permittivity as a function of the electromagnetic field. It is shown that an exact analytical solution of Maxwell's equations corresponding to the TM surface polariton in the form described by sech function do exist in a Kerr-like nonlinear medium with the permittivity profile given by a hypergeometric function. We compare our analytical solution and analogous exact numerical solution in a Kerr medium. The power flow down the interface between two media is also discussed.     

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.     

Formation of dust-acoustic shock waves in a strongly coupled cryogenic dusty plasma

The nonlinear propagation of ultra-low-frequency dust-acoustic (DA) waves in a strongly coupled cryogenic dusty plasma has been investigated, by using the Boltzmann distributed electrons and ions, as well as modified hydrodynamic equations for strongly coupled charged dust grains. The reductive perturbation technique is used to derive the Burger equation. It is shown that strong correlations among negatively charged dust particles acts like a dissipation, which is responsible for the formation of the DA shock waves. The latter are associated with the negative potential, i.e. with the compression of negatively charged cryogenic dust particle density. It is also found that the effective dust-temperature, which arises from electrostatic interactions among negatively charged dust particles, significantly affects the height of the DA shock structures. New laboratory experiments at cryogenic temperature should be conducted to verify our theoretical prediction.     

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)     

Dispersion and nonlinear frequancy shift of natural waves in the Bose–Einstein condensate

Quantum mechanics equations for a system of the Bose particles are represented in the form of material field equations. A nonlinear equation for the macroscopic one-particle wave function is derived. Using the Krylov–Bogolyubov–Mitropol’skii method for equations in partial derivatives, nonlinear waves in the Bose–Einstein condensate are investigated. In the cubic approximation, dispersion relations for waves are derived and nonlinear frequency shift is calculated in the first- and third-order approximations for the interaction radius.     

Nonlinear noise waves in soft biological tissues

The study of intense waves in soft biological tissues is necessary both for diagnostics and therapeutic aims. Tissue represents an inherited medium with frequency-dependent dissipative properties, in which waves are described by nonlinear integro-differential equations. The equations for such waves are well known. Their group analysis has been performed, and a number of exact solutions have been found. However, statistical problems for nonlinear waves in tissues have hardly been studied. As well, for medical applications, both intense noise waves and waves with fluctuating parameters can be used. In addition, statistical solutions are simpler in structure than regular solutions; they are useful for understanding the physics of processes. Below a general approach is described for solving nonlinear statistical problems applied to the considered mathematical models of biological tissues. We have calculated the dependences of the intensities of the narrowband noise harmonics on distance. For wideband noise, we have calculated the dependence of the spectral integral intensity on distance. In all cases, wave attenuation is determined both by the specific dissipative properties of the tissue and the nonlinearity of the medium.     

Quantum Field Theory of Black-Swan Events

Free and weakly interacting particles are described by a second-quantized nonlinear Schrödinger equation, or relativistic versions of it. They describe Gaussian random walks with collisions. By contrast, the fields of strongly interacting particles are governed by effective actions, whose extremum yields fractional field equations. Their particle orbits perform universal Lévy walks with heavy tails, in which rare events are much more frequent than in Gaussian random walks. Such rare events are observed in exceptionally strong windgusts, monster or rogue waves, earthquakes, and financial crashes. While earthquakes may destroy entire cities, the latter have the potential of devastating entire economies.     

Alfvénic shock waves in a collisional magnetoplasma

Compressional Alfvénic shock waves in a cold collisional magnetoplasma are investigated. For this purpose, we use the hydrodynamic equations and Faraday?s law to derive the governing nonlinear equations for the compressional Alfvén waves. It is shown that the latter can appear in the form of Alfvénic shock waves.     

Collisionless Boltzmann equations and integrable moment equations

A collisionless Boltzmann equation, describing long waves in a dense gas of particles interacting via short-range forces, is shown to be equivalent to the Benney equations, which describe long waves in a perfect two-dimensional fluid with a free surface. These equations also describe, in a random phase approximation, the evolution, on long space and time scales, of multiply periodic solutions of the nonlinear Schrödinger equation. The derivative nonlinear Schrödinger equation is likewise shown to be related to an integrable system of moment equations.     

Modelling of nonlinear surface gravity waves under shallow-water conditions with account of dispersion

The modelling of nonlinear surface gravity waves under shallow-water conditions with account of dispersion is described in this study. On the basis of the analytic expressions obtained for the horizontal velocity of medium particles, the profile evolution of nonlinear surface gravity waves during its propagation under shallow-water conditions is described. The profiles of surface gravity waves during their propagation in the bay with account of dispersion are given.     

Cylindrical electromagnetic waves in a nonlinear nondispersive medium: Exact solutions of the Maxwell equations

The excitation and propagation of cylindrical electromagnetic waves in a nonlinear nondispersive medium are analyzed. It is assumed that the medium lacks a center of symmetry and that the dependence of the electric displacement on the electric field can be approximated by an exponential function. For this case, a method for integrating the system of the Maxwell equations is proposed. Exact solutions to a set of nonlinear electromagnetic field equations are obtained by this method. It is shown that nonlinear effects described by these solutions can become apparent under experimental conditions.     

Specificity of the phenomenon of acoustoelasticity in a two-dimensional internally structured medium

A two-dimensional model of a microstructured medium is considered in the form of a square lattice consisting of elastically interacting circular particles with translational and rotational degrees of freedom. The interactions between the particles are modeled by a set of elastic springs. Differential equations are derived to describe the propagation and interaction of acoustic waves in such a medium. The relation between the velocities of wave propagation and the small strain arising in the structure under external action is determined. Analytical expressions that determine the difference between the squares of the velocities of both longitudinal and shear waves propagating in two mutually perpendicular directions in a medium with an externally induced anisotropy are derived and analyzed.     

Propagation of electromagnetic waves in a medium with nonlinear magnetic susceptibility

The interaction of electromagnetic waves in a medium with nonlinear magnetic susceptibility is investigated. On the basis of Tinkham's measurements an antiferromagnet showing a resonance in the submillimeter region is considered as a nonlinear medium. The solution of both the Landau-Lifshitz and Maxwell equations leads to the RHS wave equation. Its solution with boundary conditions shows that the amplitude of the reflected wave from the nonlinear medium on a combined frequency can be non-zero.     

Nonlinear scatterers in nonlinear media: Born approximation and local perturbation method

Wave scattering by inhomogeneous nonlinear particles placed in a nonlinear host medium is studied. In the case of weak scattering the scattering indicatrix is calculated in the first Born approximation. Scattering from small nonlinear particles loaded in a medium with saturation of permittivity is studied by the local perturbation method. A small perturbation method is developed for nonlinear equations of rather general type with a random, intensity-dependent, scattering potential.     

Wave processes in media with hysteretic nonlinearity. Part I

Two phenomenological models of hysteretic equations of state for media with imperfect elasticity are described and compared. On the basis of these equations, a theoretical study of nonlinear effects caused by the acoustic wave propagation in an unbounded medium is performed. The profiles, parameters, and spectra of waves are determined. The distinctive features of nonlinear wave processes in such media are revealed, so that these features can be used to choose the appropriate hysteretic equation of state for analytically describing the experimental data.     

Nonlinear interactions between drift waves and zonal flows

Phase coherent interactions between drift waves and zonal flows are considered. For this purpose, mode coupling equations are derived by using a two-fluid model and the guiding center drifts. The equations are then Fourier analyzed to deduce the nonlinear dispersion relations. The latter depict the excitation of zonal flows due to the ponderomotive forces of drift waves. The flute-like zonal flows with insignificant density fluctuations have faster growth rates than those which have a finite wavelength along the magnetic field direction. The relevance of our investigation to drift wave driven zonal flows in computer simulations and laboratory plasmas is discussed. Received 5 April 2002 Published online 28 June 2002