Localization of nonlinear waves in randomly inhomogeneous media (original) (raw)
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1993
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Localization properties of acoustic waves in the random-dimer media
Physical Review B, 2008
Propagation of acoustic waves in the one-dimensional (1D) random dimer (RD) medium is studied by three distinct methods. First, using the transfer-matrix method, we calculate numerically the localization length ξ of acoustic waves in a binary chain (one in which the elastic constants take on one of two values). We show that when there exists short-range correlation in the mediumwhich corresponds to the RD model -the localization-delocalization transition occurs at a resonance frequency ω c . The divergence of ξ near ω c is studied, and the critical exponents that characterize the power-law behavior of ξ near ω c are estimated for the regimes ω > ω c and ω < ω c . Second, an exact analytical analysis is carried out for the delocalization properties of the waves in the RD media. In particular, we predict the resonance frequency at which the waves can propagate in the entire chain. Finally, we develop a dynamical method, based on the direct numerical simulation of the governing equation for propagation of the waves, and study the nature of the waves that propagate in the chain. It is shown that only the resonance frequency can propagate through the 1D media. The results obtained with all the three methods are in agreement with each other.
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Physica B: Condensed Matter, 2003
An original approach to the description of classical wave localization and diffusion in random media is developed. This approach accounts explicitly for the correlation properties of the disorder and is general with respect to the dimensionality of the system. Specifically, we evaluate a two-frequency mutual coherence function, which is an important quantity in itself, and also because the spectral correlator determines the evolution of transient signals in the time domain. The predictions of our theory describe a ballistic to diffusive transition in the wave transport, and, for not too large distances (not exceeding, roughly, several hundreds of mean-free paths) are consistent, in general, with a classical diffusion paradigm. Since the coherence function is expressed via an arbitrary form power spectrum, the results obtained in the work open a new avenue in studying wave transport in anisotropic and/or fractally correlated systems. r
Localization problem and mapping of one-dimensional wave equations in random and quasiperiodic media
Physical Review B, 1986
The one-dimensional Schrixhnger equation with multiple scattering potentials is transformed to a discrete (tight-binding) form exactly. For a random configuration of potentials in which a11 the states are localized, it is shown (not argued} that the resistance p behaves as p-exp(yi) at a large distance I, where y is the Lyapunov exponent (inverse of the localization length) of corresponding transfer matrices. In a case where two (or more) types of scatterers are arranged in a quasiperiodic manner (for example, the Fibonacci series), it is shown that wave functions are always critical, namely they are either self-similar or chaotic, and are intermediate between localized and extended states.
Statistical and dynamic localization of plane waves in randomly layered media
Soviet Physics Uspekhi, 1992
This article presents a detailed discussion of the problem of the localization and various methods of describing it on the basis of plane waves multiply scattered in randomly layered media. It is noted that the field of localized waves has a complicated structure, with sharp peaks and extended "dark" regions, where the intensity of the wave is small. It is shown that because of this complicated structure the wave field in a randomly layered medium, the dynamic and statistical characteristics of the wave behave in fundamentally different ways. For example, the statistical moments of the intensity of the wave increase exponentially into the interior of the medium, while the energy of the wave penetrating into randomly inhomogeneous medium can be finite with unity probability. The concept of a majorant curve and of an isoprobability curve, helpful for understanding the phenomenon of localization, are introduced. Also taken into account is the effect of a small regular absorption on the statistical and dynamic properties of the wave, and the localization of space-time pulses in a randomly layered medium is also studied.
Transient Waves in a Class of Random Heterogeneous Media
Applied Mechanics Reviews, 1991
A stochastic method is developed for analysis of transient waves propagating in one-dimensional randoai granular-type media. The method is suited to study transient dynamic response of nonlinear microstructures with material randomess, of high signal-to-noise ratio, being present in constitutive moduli and grain lengths. It generalizes the classical solution techniques, based on the theory of characteristics, by taking advantage of the Markov property of the forward propagaing disturbances. Pulses propagating in bilinear elastic, nonlinear elastic, and linear-hysteretic media are studied. Additionally, a short review is given of an investigation of acceleration wavefronts making a transition into shocks in random nonlinear elastic/dissipative continua, where Markov property can again be exploited.
Localized waves in inhomogeneous media
Soviet Physics Uspekhi, 1984
This review examines the localization of one-dimensional nonlinear waves in an inhomogeneous multiphase medium. Particular attention is devoted to the localization of two types of waves, namely, solitary waves (domains) and switching waves that are the separation boundaries between the corresponding phases (domain walls). The localized state of such waves on both point and slowly-varying (in space) inhomogeneities is investigated. It is shown that several types of waves can become localized on inhomogeneities, and variation of external parameters may be accompanied by abrupt transitions between different types of localized waves. The stability of waves localized on inhomogeneities is examined together with various hysteresis phenomena that may occur in an inhomogeneous medium. The general results presented in the first part of the review are illustrated by examples of different physical systems, including superconductors, normal metals, semiconductors, plasmas, and chemical-reaction waves.
Physical Review E, 1997
We study numerically the evolution of wavepackets in quasi one-dimensional random systems described by a tight-binding Hamiltonian with long-range random interactions. Results are presented for the scaling properties of the width of packets in three time regimes: ballistic, diffusive and localized. Particular attention is given to the fluctuations of packet widths in both the diffusive and localized regime. Scaling properties of the steady-state distribution are also analyzed and compared with a theoretical expression borrowed from the one-dimensional Anderson theory. Analogies and differences with the kicked rotator model and the one-dimensional localization are discussed.
Stochastic differential equation approach for waves in a random medium
Physical Review E, 2009
We present a mathematical approach that simplifies the theoretical treatment of electromagnetic localization in random media and leads to closed form analytical solutions. Starting with the assumption that the dielectric permittivity of the medium has delta-correlated spatial fluctuations, and using Ito's lemma, we derive a linear stochastic differential equation for a one dimensional random medium. The equation leads to localized wave solutions. The localized wave solutions have a localization length that scales as 2 −