On the existence of non-polarized azimuthal-symmetric electromagnetic waves in circular dielectric waveguide filled with nonlinear isotropic homogeneous medium

The propagation of monochromatic nonlinear symmetric hybrid waves in a cylindrical nonlinear dielectric waveguide is considered. The physical problem is reduced to solving a transmission eigenvalue problem for a system of ordinary differential equations. Spectral parameters of the problem are propagation constants of the waveguide. The problem is reduced to the new type of nonlinear eigenvalue problem. The analytical method of solving this problem is presented. New propagation regime is found.     

Properties of nonlinear waves in an actively-dissipative dispersive medium

The wave processes in an actively-dissipative dispersive medium described by a nonlinear evolutionary fourth-order equation are considered. With the use of traveling-wave variables analytical solutions in the form of solitary waves and kinks are obtained for certain combinations of the problem parameters. The stability of the exact solutions obtained is studied. The processes of formation of stable periodic oscillations are considered for different model parameters. The control parameter ranges, on which periodic structures can be formed, are determined.     

Scattering and depolarization of electromagnetic waves by a horizontally weakly inhomogeneous medium

A general theory of wave scattering from a weakly inhomogeneous medium is developed for the case where the inhomogeneity varies parallel to the boundary plane. The method of small perturbation is used and terms are carried up to and including the second order. It is found that the scattered waves are depolarized and present in all directions. In the special case of forward- or backscattering the depolarized fields are of the second order and are seen to result from a multiple scattering process; while in other directions, these fields could be of the first order, and result from a single scattering process.     

Higher-order approximations in the analysis of nonlinear cylindrical waves in a hyperelastic medium

The nonlinear wave equation is solved analytically in cylindrical coordinates using the third-order approximation of the Hankel function. The second-order and third-order solutions are compared. The evolution of the initial wave profile is simulated numerically for different initial frequencies __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 36–45, April 2007.     

Numerical detection and generation of solitary waves for a nonlinear wave equation

This paper presents an automatic algorithm for detecting and generating solitary waves of nonlinear wave equations. With this purpose, dynamic simulations are carried out, the solution of which evolves into a main pulse along with smaller dispersive tails. The solitary waves are detected automatically by the algorithm by checking that they have constant amplitude and are symmetric respect to its maximum value. Once the main wave has been detected, the algorithm cleans the dispersive tails for time enough so that the solitary wave is obtained with the required precision.In order to use our algorithm, we need a spatial discretization with local basis. The numerical experiments are carried out for the BBM equation discretized in space with cubic finite elements along with periodic boundary conditions. Moreover, a geometric integrator in time is used in order to obtain good approximations of the solitary waves.     

Propagation of nonlinear waves in a porous medium with two-phase saturation by a liquid and a gas

The propagation of nonlinear waves through a porous medium saturated with a viscous liquid and a gas is investigated with allowance for the capillary pressure. Numerical solutions of the traveling-wave type are constructed for the generalized Korteweg-de Vries-Burgers equation for the wave amplitudes. Three types of regimes of longitudinal wave propagation, including soliton-like regimes, are found.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 86–95, July–August, 1996.     

Sound propagation and self-focusing in an inhomogeneous gas-liquid medium

The evolution of sound waves in a gas—liquid medium with an inhomogeneous distribution of the sound speed is considered in this paper on the basis of a nonlinear parabolic equation for the amplitude of the sound wave envelope. It is assumed that the nonlinearity due to the gas inclusions is much greater than the customary hydrodynamic nonlinearity. The influence of the inhomogeneity on the self-focusing of the sound is studied.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 177–180, November–December, 1974.The author is grateful to I. R. Shreiber for discussion.     

Sound waves in a magnetizable medium

This article considers the propagation of small perturbations in a medium which can be inhomogeneously and isotropically magnetized under the action of an electromagnetic field. It is shown that in such a medium there is the possibility of sound waves of the same kind as in a medium with a constant magnetic susceptibility. However, the phase velocities of fast and slow magnetosonic waves can take on imaginary values so that, in strong magnetic fields, there may arise the phenomenon of instability. Investigations were made of the diagrams of the phase velocities for para- and diamagnetic substances for a medium with magnetic saturation; the case of an incompressible medium is discussed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 15–22, January–February, 1973.     

Numerical study of formation of solitary strain waves in a nonlinear elastic layered composite material

Propagation of nonlinear strain waves through a layered composite material is considered. The governing macroscopic wave equation for the long-wave case was obtained earlier by the higher-order asymptotic homogenization method (Andrianov et al., 2013). Non-stationary dynamic processes are investigated by a pseudo-spectral numerical procedure. The time integration is performed by the Runge–Kutta method; the approximation with respect to the spatial co-ordinate is provided by the Fourier series expansion. The convergence of the Fourier series is substantially improved and the Gibbs–Wilbraham phenomenon is reduced with the help of Padé approximants. As result, we explore how fast and under what conditions the solitary strain waves can be generated from an initial excitation. The numerical and analytical solutions (when the latter can be obtained) are in good agreement.     

Reflection and transmission of electromagnetic waves from a nonlinear anisotropic slab between two linear isotropic media

Summary In this paper, the author has studied the reflection from and transmission through a homogenous nonlinear anisotropic slab (anisotropy being due to an external magnetic field) bounded by two linear isotropic media. Nonlinear equations describing the growth of the two modes of propagation of an electromagnetic wave in the direction of the magnetic field, in an anisotropic nonlinear medium, have been set up and solved; the solutions have been used to obtain expressions for the reflected and transmitted components of the incident wave. The simpler problem of the reflection and transmission from an isotropic nonlinear slab has also been discussed as a special case.