Wave propagation in nonlinear disordered media

We analyze mechanisms and regimes of wave packet spreading in nonlinear disordered media. We discuss resonance probabilities, and predict a dynamical crossover from strong to weak chaos. The crossover is controlled by the ratio of nonlinear frequency shifts and the average eigenvalue spacing of eigenstates of the linear equations within one localization volume. We consider generalized models in higher lattice dimensions and obtain critical values for the nonlinearity power, the dimension, and norm density, which influence possible dynamical outcomes in a qualitative way.     

Thermal conductivity of nonlinear waves in disordered chains

We present computational data on the thermal conductivity of nonlinear waves in disordered chains. Disorder induces Anderson localization for linear waves and results in a vanishing conductivity. Cubic nonlinearity restores normal conductivity, but with a strongly temperature-dependent conductivity κ(T). We find indications for an asymptotic low-temperature κ ∼ T ⁴ and intermediate temperature κ ∼ T ² laws. These findings are in accord with theoretical studies of wave packet spreading, where a regime of strong chaos is found to be intermediate, followed by an asymptotic regime of weak chaos (Laptyeva et al, Europhys. Lett. 91, 30001 (2010)).     

Nonlinear coherent transport of waves in disordered media

We present a diagrammatic theory for coherent backscattering from disordered dilute media in the nonlinear regime. We show that the coherent backscattering enhancement factor is strongly affected by the nonlinearity, and we corroborate these results by numerical simulations. Our theory can be applied to several physical scenarios such as scattering of light in a nonlinear Kerr medium or propagation of matter waves in disordered potentials.     

TM-polarized nonlinear slab-guided waves in saturable media

The power-dependent wave vector is calculated for TM-polarized nonlinear waves guided by thin dielectric film bounded on one side by a nonlinear saturable medium. The results are compared with those obtained for Kerr-like nonlinear media.     

Stability of solitary waves in nonlinear composite media

The system of equations for planar waves in elastic composite media in the presence of anisotropy is considered. In anisotropic case two two-parametric families of solitary waves are found in an explicit form. In case of the absence of anisotropy these two families coalesce into the unique three parametric family. The solitary wave solutions are found to be orbitally stable in a certain range of their phase speeds (range of stability) both in an anisotropic as well as in an isotropic materials. It is also shown that the initial value problem for the governing equations is locally well posed which is needed to prove the stability result. The local well-posedness of the initial value problem along with stability of solitary waves implies global existence result provided the initial data lie in a neighbourhood of a stable solitary wave. This complements the previous results of blow-up for this type of equations.     

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.     

Anderson localization and propagation of electromagnetic waves through disordered media

We have used the dynamic method to calculate the frequency dependence of the localization length in a disordered medium, using the amplitude change and the redshift of the spectral density of the propagating incident pulse. The frequency dependence of the localization length in an effectively one-dimensional disordered medium is computed in terms of the strength of the disorder. The results obtained with the dynamic method are confirmed by computing the same results using the transfer-matrix method.     

Highly nonlinear solitary waves in heterogeneous periodic granular media

We use experiments, numerical simulations, and theoretical analysis to investigate the propagation of highly nonlinear solitary waves in periodic arrangements of dimer (two-mass) and trimer (three-mass) cell structures in one-dimensional granular lattices. To vary the composition of the fundamental periodic units in the granular chains, we utilize beads of different materials (stainless steel, brass, glass, nylon, polytetrafluoroethylene, and rubber). This selection allows us to tailor the response of the system based on the masses, Poisson ratios, and elastic moduli of the components. For example, we examine dimer configurations with two types of heavy particles, two types of light particles, and alternating light and heavy particles. Employing a model with Hertzian interactions between adjacent beads, we find good agreement between experiments and numerical simulations. We also find good agreement between these results and a theoretical analysis of the model in the long-wavelength regime that we derive for heterogeneous environments (dimer chains) and general bead interactions. Our analysis encompasses previously-studied examples as special cases and also provides key insights on the influence of heterogeneous lattices on the properties (width and propagation speed) of the nonlinear wave solutions of this system.     

On the second-type nonlinear elastic waves in porous media

Nonlinear second-type (matrix) waves are studied with special emphasis on the formation of saw-tooth shock waves. Configurations of the elastic waves in specific cases of porous gas-saturated sedimentary are calculated.     

Doppler effect of nonlinear waves and superspirals in oscillatory media

Nonlinear waves emitted from a moving source are studied. A meandering spiral in a reaction-diffusion medium provides an example in which waves originate from a source exhibiting a back-and-forth movement in a radial direction. The periodic motion of the source induces a Doppler effect that causes a modulation in wavelength and amplitude of the waves ("superspiral"). Using direct simulations as well as numerical nonlinear analysis within the complex Ginzburg-Landau equation, we show that waves subject to a convective Eckhaus instability can exhibit monotonic growth or decay as well as saturation of these modulations depending on the perturbation frequency. Our findings elucidate recent experimental observations concerning superspirals and their decay to spatiotemporal chaos.     

Stability and instability of nonlinear guided waves in saturable media

The stability of stationary TE_m thin film-guided waves in asymmetric planar waveguides with a self-focusing saturable nonlinear cladding was investigated numerically. The results indicate that the TE₀ wave is unstable to propagation on the negatively sloped branch of the nonlinear dispersion curve.     

Self-similar counter-propagating beams of electromagnetic waves in nonlinear cubic media

We study the structure of the self-similar beams of counter-propagating electromagnetic waves in nonlinear cubic media. Eigenfrequencies of two-mirror quasi-optical resonators, whose eigen-modes are self-similar beams of counter-propagating waves, are found.