Nonlinear transparency regimes for three-component acoustic pulses in a system of electron and nuclear spins

We study acoustic solitons consisting of one longitudinal and two transverse components and propagating in the direction perpendicular to an external magnetic field in a crystal containing paramagnetic impurities of electron and nuclear spins. The coupling of the electron spin subsystem to the longitudinal sound allows making the velocity of the latter close to that of the transverse acoustic waves, which provides an effective interaction between all components of the elastic field by means of the nuclear spin subsystem. We derive a three-component system of material and reduced wave equations describing this process and construct its soliton solutions in the form of stationary and breather pulses. Based on them, we study the peculiarities of the dynamics of the elastic field components and reveal the differences from the two-component model. The existence of two families of breathers is an important distinctive feature of the considered case.     

Integrable models of the dynamics of longitudinal-transverse acoustic pulses in a paramagnetic crystal

We study the interaction between longitudinal-transverse acoustic pulses and a system of paramagnetic impurities with the effective spin S = 1 in a statically deformed crystal. We show that the dynamics of a pulse propagating at an arbitrary angle to the static-deformation direction and of the effective spins satisfy the modified reduced Maxwell-Bloch equations and, if the spectrum of the acoustic pulse overlaps the quantum transitions between spin sublevels, the modified sine-Gordon equation. These equations generalize the well-known models in the theory of the inverse scattering method and in the theory of self-induced transparency and also belong to the class of integrable equations. Analyzing soliton solutions shows that the pulse-medium interaction reveals some qualitatively new features in these models compared with the cases of purely transverse or purely longitudinal acoustic fields. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 228–247, May, 2007.     

弹性固体和充满两种互不相溶粘性流体的多孔固体之间界面的反射和折射及其衰减波

在充满两种互不相溶粘性流体的多孔固体中,研究弹性波的传播.用3个数性的势函数描述3个纵波的传播,用1个矢性的势函数单独描述横波的传播.根据这些势函数,在不同的组合相中,定义出质点的位移.可以看出,可能存在3个纵波和1个横波.在一个弹性固体半空间与一个充满两种互不相溶粘性流体的多孔固体半空间之间,研究其界面上入射纵波和横波所引起的反射和折射现象.由于孔隙流体中有粘性,折射到多孔介质中的波,朝垂直界面方向偏离.将入射波引起的反射波和折射波的波幅比,作为非奇异的线性代数方程组计算.进一步通过这些波幅比,计算出各个被离散波在入射波能量中所占的份额.通过一个特殊的数值模型,计算出波幅比和能量比系数随入射角的变化.超过SV波的临界入射角,反射波P将不再出现.越过界面的能量守恒原理得到了验证.绘出了图形并对不同孔隙饱和度以及频率的变化,讨论它们对能量分配的影响.     

Interaction of Elastic Waves

We derive and analyze asymptotic equations for the interaction of weakly nonlinear elastic waves. We show that there are resonant triads involving two transverse and one longitudinal wave provided the wave speeds satisfy a certain irrationality condition. We study initial value and signaling problems, and the interaction of sawtooth wave packets.     

A method of integral equations in nonlinear boundary-value problems for flat shells of the Timoshenko type with free edges

We prove the existence theorem for solutions of geometrically nonlinear boundary-value problems for elastic shallow isotropic homogeneous shells with free edges under shear model of S. P. Timoshenko. Research method consists in the reduction of the original system of equilibrium equations to a single nonlinear equation for the components of transverse shear deformations. The basis of this method are integral representations for the generalized displacements, containing an arbitrary holomorphic functions, which are determined by the boundary conditions involving the theory of one-dimensional singular integral equations.     

Waves and vibrations in strings with windings: Commemorating the 100th anniversary of the birth of Kh.A. Rakhmatulin

The equations of the propagation of transverse, twisting and longitudinal waves and vibrations are obtained, taking into account their interactions in musical strings with windings. Their solutions are obtained. The occurrence of transverse and twisting motions leads to the appearance of longitudinal motions, while the transverse and longitudinal components play the role of inducing forces for the twisting components. The contributions of the transverse, twisting and longitudinal components to the dynamic loading of the string are of the same order. The longitudinal-twisting vibrations occur both at natural frequencies and at frequencies of the transverse vibrations. Resonance phenomena between the individual modes of these vibrations are possible.     

具强迫项非线性梁方程解的渐近性

本文同时考虑纵横弯曲及粘性效应,建立了一类轴向载荷和横向载荷作用下的非线性粘弹性简支梁方程。利用Faedo-Galerkin法,证明了该方程解的存在唯一性,讨论了该方程解的渐近性,并给出了有界吸收集的存在性证明。     

Deformation of laminated anisotropic bars in the three-dimensional statement 1. Transverse-longitudinal bending and edge compatibility condition

By using the methods of asymptotic splitting, the 3D problem on the transverse-longitudinal bending of elastic anisotropic laminated bars under the action of longitudinal and transverse loads is split into a number of plane boundary-value problems, whose solution allows one to construct asymptotic approximations for all components of the displacement vector and the strain tensor at each point in the bars. A system of ordinary differential equations of transverse-longitudinal bending is obtained for laminated bars with arbitrary anisotropy of its layers. A condition for the edge compatibility of materials in laminated orthotropic bars is deduced. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 3, pp. 379-410, May-June, 2009.     

Two-dimensional wave propagation in a generalized elastic solid

The object of the present study is to investigate the propagation of two-dimensional waves in a weakly nonlinear and weakly dispersive elastic solid. The reductive perturbation method is directly applied to a Lagrangian whose Euler–Lagrange equations give the field equations for a quadratically nonlinear elastic medium with higher order gradients. In the long-wave approximation, it is shown that the long-time behavior of the two transverse waves is governed by the two coupled modified Kadomtsev–Petviashvili (CMKP) equations. Depending on the choice of the direction of perpendicular dynamics, various forms of the CMKP equations are obtained. Some special solutions are also presented for a simplified form of the CMKP equations.     

Propagating waves in elastic material with voids subjected to electro-magnetic interactions

Constitutive relations and field equations are developed for an elastic solid with voids subjected to electro-magnetic field. The linearized form of the relations and equations are presented separately when medium is subjected to a large magnetic field and when it is subjected to a large electric field. The possibility of propagation of time harmonic plane waves in an infinite elastic solid with voids has been explored. It is found that when the medium is subjected to large magnetic field, there exist two coupled longitudinal waves propagating with distinct speeds and a transverse wave mode. However, when the medium is subjected to a large electric field, there may propagate five basic waves comprising of four coupled longitudinal waves propagating with distinct speeds and a lone transverse wave. The effects of magnetic and electric fields are observed on the propagation characteristics of the existing waves. Under the limiting cases of frequency and for different electric conductive materials, the speeds of various waves are investigated. The phase speeds of different waves and their corresponding attenuations have been computed against the frequency parameter and depicted graphically for a specific material.     

Novel rogue waves in an inhomogenous nonlinear medium with external potentials

Employing the similarity transformation connected with the standard constant coefficient nonlinear Schrödinger equation, we obtain the analytical rogue wave solutions to a generalized variable coefficient nonlinear Schrödinger equation with external potentials describing the pulse propagation in nonlinear media with transverse and longitudinal directions nonuniformly distributed. Based on the obtained solutions, abundant structures of rogue waves are constructed by selecting some special parameters. The main properties as well as the dynamic behaviors of these rogue waves are discussed by direct computer simulations.     

Asymptotic expansion of the solution to the system of elasticity equations with a concentrated pulse force

For the linear system of elasticity equations, we consider the problem of wave excitation by a concentrated pulse force of arbitrary orientation. On assuming that the medium is isotropic and its density and elastic moduli are infinitely differentiable functions constant in some neighborhood of the source point, we write down an asymptotic series for the solution. The coefficients of the series determine the singular component of the solution, as well as the jumps, of the solution and its derivatives, as the characteristic cones are crossed, corresponding to longitudinal and transverse waves.     

Transverse linear instability of solitary waves for coupled long-wave–short-wave interaction equations

In this paper, we investigate the transverse linear instability of one-dimensional solitary wave solutions of the coupled system of two-dimensional long-wave–short-wave interaction equations. We show that the one-dimensional solitary waves are linearly unstable to perturbations in the transverse direction if the coefficient of the term associated with transverse effects is negative. This transverse instability condition coincides with the non-existence condition identified in the literature for two-dimensional localized solitary wave solutions of the coupled system.     

An asymptotic approach to the problem of the free oscillations of a beam

Equations describing the free small longitudinal and transverse oscillations of a straight elastic beam of rectangular cross section are obtained using the plane linear theory of elasticity and the method of integrodifferential relations. The initial system of partial differential equations is reduced to a system of ordinary linear differential equations with constant coefficients. The effect of the geometrical and elastic characteristics of the beam on the frequency and form of the natural oscillations is investigated. For longitudinal motions it is shown that different types of natural displacements and internal stresses of the beam exist. For transverse oscillations, it is found that there are frequency zones corresponding to different forms of the solutions of the characteristic equation obtained using the proposed model.     

Coherent Structures in Weakly Birefringent Nonlinear Optical Fibers

This article studies two coupled nonlinear Schrodinger equations that govern the pulse propagation in weakly birefringent nonlinear optical fibers. The coherent structures for these equations, such as vector solitons and localized oscillating solutions, are studied analytically and numerically. Three types of localized oscillating structures are identified and their functional forms determined by perturbation methods. In some of these structures, infinite oscillating tails are present. The implications of these tails are also discussed.     

Eigenwaves in a relativistic heat-conducting viscous fluid according to extended irreversible thermodynamics

We propose an approach to extended irreversible thermodynamics leading to a set of transport equations for the dissipative variables characterizing a relativistic heat-conducting viscous fluid. After this, we study the propagation eigenmodes showing the possible existence of three non-trivial modes: two longitudinal and one transverse. The fast longitudinal wave is a pressure wave associated to a slow longitudinal thermal dissipation wave and a transverse viscous dissipation wave. The general study of the longitudinal case is not possible. Natural special cases are analyzed in detail.     

Localized asymptotic solutions of the navier-stokes equations and laminar wakes in an incompressible fluid

Equations are derived to describe the far-field laminar wake behind a body in incompressible fluid flow with an arbitrary distribution of the free-stream (unperturbed flow) velocity. For certain classes of free-stream flows, analysis of these equations enables various processes in narrow wakes or jets to be described (the interaction of the longitudinal transverse velocity components in a jet, cause it to accelerate or decelerate and conservation of the energy of the wake by distortion of its trajectory regardless of viscous dissipation). In particular, conditions are obtained for the wake growth in spiral flows, analogous to the Rayleigh conditions for the instability of two-dimensionally radially symmetric flows relative to three-dimensional short-wave perturbations.     

Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control

We consider systems of Timoshenko type in a one-dimensional bounded domain. The physical system is damped by a single feedback force, only in the equation for the rotation angle, no direct damping is applied on the equation for the transverse displacement of the beam. Moreover the damping is assumed to be nonlinear with no growth assumption at the origin, which allows very weak damping. We establish a general semi-explicit formula for the decay rate of the energy at infinity in the case of the same speed of propagation in the two equations of the system. We prove polynomial decay in the case of different speed of propagation for both linear and nonlinear globally Lipschitz feedbacks.      

Waves of polarized light in nonlinear birefringent fibre

A propagation of a short optical pulse in nonlinear birefringent fibre is described by a system of two coupled Schrödinger equations. By means of variational Anderson method this system reduces to the system of ordinary differential equations for spatial evolution of pulse parameters. In two ultimate cases the analytical solutions of the equations are managed to be found. It is shown that at some critical power of the input pulse W_c the regime of propagation changes. For the power exceeding W_c the radiation concentrates in one channel. The numerical investigation of the intermediate cases was done when by the variation of the input pulse power one can achieve the comparable effectiveness of the competing processes of dispersion broadening and nonlinear pulse compression. The numerical simulations show that in the range of critical values of the nonlinear coupling coefficient the transition takes place to the chaotic phase and amplitude behavior of the coupled waves of different polarizations. The research is important to understand the processes of ultra short digital pulses propagation in optical fibre links.