Energy propagation for dispersive waves in dissipative media

University of Bologna, Italy. Published in Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 7, pp. 650–664, July, 1993.     

Nonlinear waves in weakly dispersive random media

The propagation of nonlinear waves in random media is an important aspect of nonlinear wave theory and has a long and informative history. This paper describes the basic ideas of the approaches that have been applied. The average-field method, which has been used most extensively in linear problems, is considered. This approach is then shown to be incorrect as far as nonlinear processes are concerned. Finally, a new scheme is proposed average-form the method, which allows consistent evolution equations to be obtained for nonlinear waves in random media.Institute of Applied Physics, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 8, pp. 760–766, August, 1993.     

Breaking of stationary waves in nonlinear dispersive media

A nontrivial behaviour of a nonlinear wave under influence of small disturbing factors like dissipation, smooth inhomogeneity of medium parameters, etc. is considered by the example of sine-Gordon equation. The stage of slow “adiabatic” variation of the parameters of quasi-stationary wave is shown to change at some finite distance due to strong instability. The wave form becomes essentially nonstationary (breaking of stationary wave structure). The breaking condition is defined by the extremum of the wave adiabatic invariant. The behaviour of a wave at the nonadiabatic stage is described using a Galerkin procedure.     

Hydrodynamics of modulated finite-amplitude waves in dispersive media

Analytic approaches are developed for integrating the nondiagonalizable Whitham equations for the generation and propagation of nonlinear modulated finite-amplitude waves in dissipationless dispersive media. Natural matching conditions for these equations are stated in a general form analogous to the Gurevich-Pitaevskii conditions for the averaged Korteweg-de Vries equations. Exact relationships between the hydrodynamic quantities on different sides of a dissipationless shock wave, an analog of the shock adiabat in ordinary dissipative hydrodynamics and first proposed on the basis of physical considerations by Gurevich and Meshcherkin, are obtained. The boundaries of a self similar, dissipationless shock wave are determined analytically as a function of the density jump. Some specific examples are considered. Zh. éksp. Teor. Fiz. 115, 1116–1136 (March 1999)     

Singularities of the Hele-Shaw flow and shock waves in dispersive media

We show that singularities developed in the Hele-Shaw problem have a structure identical to shock waves in dissipativeless dispersive media. We propose an experimental setup where the cell is permeable to a nonviscous fluid and study continuation of the flow through singularities. We show that a singular flow in this nontraditional cell is described by the Whitham equations identical to Gurevich-Pitaevski solution for a regularization of shock waves in Korteveg-de Vriez equation. This solution describes regularization of singularities through creation of disconnected bubbles.     

分层介质半空间瑞利波的时频分析

对分层介质半空间瑞利波的频散特性,用一种时频分析方法——重排的平滑伪魏格纳维尔分布(RSPWVD,Reassignment of Smooth Pseudo-Wigner Ville Distribution)进行了分析和研究。对均匀半空间和两层介质半空间的理论和实验研究表明,由于层状介质中瑞利波的频散曲线存在多个模式,所获得的群速度频散曲线在不同的频段显示出来的模式是对应表面位移幅度占主导作用的模式。频散曲线的这种模式判定对利用层状半空间的瑞利波反演介质参数是必须预先了解的。     

Treatment of Čerenkov Radiation from electric and magnetic charges in dispersive and dissipative media

Summary A rigorous treatment of the problem of Čerenkov radiation from fast-moving electric and magnetic charges is presented. This is based on the rigorous solution of Maxwell's equations in a general dispersive medium possessing dielectric and magnetic properties and with, and without, dissipation. It is shown that the fields are completely determined by one scalar function. Expressions for the exact fields are obtained. From the asymptotic fields all the relevant properties of Čerenkov radiation are reproduced. In particular, it is shown that, in the absence of dissipation, the energy in each mode travels with the phase velocity of that mode. For a dissipative medium the electric field develops a longitudinal component and the energy propagates at an angle to the phase velocity. Two experiments are suggested for the experimental detection of magnetic charges and electric-dipole moments of elementary particles based upon the Čerenkov radiation which they would emit in dispersive media.

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Riassunto Si presenta un trattamento rigoroso del problema della radiazione di Čerenkov emessa da cariche elettriche e magnetiche in moto rapido. Questo trattamento è basato sulla soluzione rigorosa delle equazioni di maxwell in un generico mezzo dispersivo avente proprietà elettriche e magnetiche con e senza dissipazione. Si dimostra che i campi sono determinati completamente da una funzione scalare. Si ottengono espressioni per i campi esatti. Dai campi asintotici sono riprodotte tutte le piú importanti proprietà della radiazione di Čerenkov. In particolare, si dimostra che, in assenza di dissipazione, l'energia in ciascun modo viaggia con la stessa velocità di fase di quel modo. Per un mezzo dissipativo il campo elettrico sviluppa una componente longitudinale e l'energia si propaga con un angolo rispetto alla velocità di fase. Si suggeriscono due esperimenti per la rivelzione sperimentale delle cariche magnetiche e dei momenti di dipolo elettrico delle particelle elementari basati sulla radiazione di Čerenkov che esse dovrebbero emettere nei mezzi dispersivi.

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Резюме Предлагается строгое рассмотрение проблемы излучения Черенкова от быстрых злектрических и магнитных зарядов. Рассмотрение основано на сторгом решении уравнений Максвелла в диспергирующей среде, обладающей дизлектрическими и магнитными свойствами, с и без диссипации. Показывается, что выражения для полей. Асимптотика полей воспроизводит все свойства излучения Черенкова. В частности, показывается, что при отсутствии диссипации знергия в каждой моде распространяется с фазовой скоростью зтой моды. Для диссипативной среды злектрическое поле проявляет продольную компоненту и знергия распространяется под углом к фазовой скорости. Предлагаются два зксперимента для зкспериментального определения магнитных зарядов и злектрических дипольных моментов злементарных частиц, основанные на излучении Черенкова, которые испускают зти частицы в диспергиующей среде.

     

Novel exact self-similar solitary waves in graded-index media with Kerr nonlinearity

A broad class of exact self-similar solitary wave solutions to the nonlinear Schrdinger equation (NLSE) are found by using the extended mapping deformation method. These novel waves can propagate inside planar, graded-index nonlinear waveguide amplifiers with self-focusing and self-defocusing Kerr nonlinearities, respectively. Furthermore, we analyze the features of the self-similar waves and their evolution. Our results in this paper include some in the literature [S.A. Ponomarenko, G.P. Agrawal, Phys. Rev. Lett. 97 (2006) 013901].     

Propagation of electromagnetic waves generated by moving sources in dispersive media

The propagation of electromagnetic waves issued by modulated moving sources of the form j( t,x ) = a( t )e ^{- iw₀ t} [(x)\dot]₀ ( t )d( x - x₀ ( t ) )j\left( {t,x} \right) = a\left( t \right)e^{ - i\omega _0 t} \dot x_0 \left( t \right)\delta \left( {x - x_0 \left( t \right)} \right) is considered, where j(t, x) stands for the current density vector, x = (x ₁, x ₂, x ₃) ∈ ℝ³ for the space variables, t ∈ ℝ for time, t → x ₀(t) ∈ ℝ³ for the vector function defining the motion of the source, ω ₀ for the eigenfrequency of the source, a(t) for a narrow-band amplitude, and δ for the standard δ function. Suppose that the media under consideration are dispersive. This means that the electric and magnetic permittivity ɛ(ω), μ(ω) depends on the frequency ω. We obtain a representation of electromagnetic fields in the form of time-frequency oscillating integrals whose phase contains a large parameter λ > 0 characterizing the slowness of the change of the amplitude a(t) and the velocity [(x)\dot]₀ ( t )\dot x_0 \left( t \right) and a large distance between positions of the source and the receiver. Applying the two-dimensional stationary phase method to the integrals, we obtain explicit formulas for the electromagnetic field and for the Doppler effects. As an application of our approach, we consider the propagation of electromagnetic waves produced by moving source in a cold nonmagnetized plasma and the Cherenkov radiation in dispersive media.     

Negative refraction and focusing of circularly polarized waves in optically active media

Analysis indicates that certain types of optically active media are capable of producing negative refraction and focusing of circularly polarized waves. It is established that a slab of such material acts just as Veselago's hypothetical left-handed media lens, providing subwavelength resolution as Sir Pendry's ideal lens, but for circularly polarized waves.     

Nonlinear ion acoustic waves in dissipative and dispersive magneto-rotating relativistic plasmas with two temperature superthermal electrons

A study has been presented for the nonlinear features of ion-acoustic (IA) shock waves in a magnetorotating plasma consisting of warm viscous streaming ions along with kappa-distributed electrons having two different temperatures. In this regard, we have employed the reductive perturbation technique to derive the Zakharov-Kuznetsov-Burgers (ZKB) equation that governs the dynamics of IA shock waves. The solution obtained by the hyperbolic tangent method has been shown to depend on various plasma parameters such as spectral index (κ_c), density fraction (f), effective rotation frequency (Ω_c), ion kinematic viscosity (η_o), and temperature ratio (σ). In the limiting case when dissipative coefficient D → 0 , we have also examined the solitary potential distributions, which are the solutions of Zakharov Kuznetsov (ZK) equation. It is found that both rarefactive and compressive structures exist for the system under consideration. The transition in the nature of such profiles is due to the enhancement in the density of cold electrons. The importance of present theoretical investigations has been carried out with regard to Saturn's magnetosphere, where two temperature superthermal electron populations have been observed by various satellite missions.     

Observation of shock transverse waves in elastic media

We report the first experimental observation of a shock transverse wave propagating in an elastic medium. This observation was possible because the propagation medium, a soft solid, allows one to reach a very high Mach number. In this extreme configuration, the shock formation is observed over a distance of less than a few wavelengths, thanks to a prototype of an ultrafast scanner (that acquires 5000 frames per second). A comparison of these new experimental data with theoretical predictions, based on a modified Burger's equation, shows good agreement.     

Induced focusing due to counter-propagation of a pair of bright and dark optical beams in self-defocusing Kerr media

We present two-dimensional numerical simulations of a nonlinear optical system made of a bright probe and a dark pump beam that counter propagate in a thin slice of self-defocusing Kerr media, simultaneously. The numerical results show that although the medium is self-defocusing to both pump and probe beam, a profile compression occurs under some conditions, and the weak bright probe beam is focused due to cross-phase modulation. The influence of the characteristics of beams and medium length on the probe beam compression is also investigated.