Effects of nonlinearity and a new nonlinear resonance in two-path phonon transmittance in lattices with two-dimensional arrays of atomic defects.

The paper is devoted to analytical and numerical studies of the effects of nonlinearity on the two-path phonon interference in the transmission through two-dimensional arrays of atomic defects embedded in a lattice. The emergence of transmission antiresonance (transmission node) in the two-path system is demonstrated for the few-particle nanostructures, which allow us to model both linear and nonlinear phonon transmission antiresonances. The universality of destructive-interference origin of transmission antiresonances of waves of different nature, such as phonons, photons, and electrons, in two-path nanostructures and metamaterials is emphasized. Generation of the higher harmonics as a result of the interaction of lattice waves with nonlinear two-path atomic defects is considered, and the full system of nonlinear algebraic equations is obtained to describe the transmission through nonlinear two-path atomic defects with an account for the generation of second and third harmonics. Expressions for the coefficients of lattice energy transmission through and reflection from the embedded nonlinear atomic systems are derived. It is shown that the quartic interatomic nonlinearity shifts the antiresonance frequency in the direction determined by the sign of the nonlinear coefficient and enhances in general the transmission of high-frequency phonons due to third harmonic generation and propagation. The effects of the quartic nonlinearity on phonon transmission are described for the two-path atomic defects with a different topology. Transmission through the nonlinear two-path atomic defects is also modeled with the simulation of the phonon wave packet, for which the proper amplitude normalization is proposed and implemented. It is shown that the cubic interatomic nonlinearity red shifts in general the antiresonance frequency for longitudinal phonons independently of the sign of the nonlinear coefficient, and the equilibrium interatomic distances (bond lengths) in the atomic defects are changed by the incident phonon due to cubic interatomic nonlinearity. For longitudinal phonons incident on a system with the cubic nonlinearity, the new narrow transmission resonance on the background of a broad antiresonance is predicted to emerge, which we relate to the opening of the additional transmission channel for the phonon second harmonic through the nonlinear defect atoms. Conditions of the existence of the new nonlinear transmission resonance are determined and demonstrated for different two-path nonlinear atomic defects. A two-dimensional array of embedded three-path defects with an additional weak transmission channel, in which a linear analog of the nonlinear narrow transmission resonance on the background of a broad antiresonance is realized, is proposed and modeled. The presented results provide better understanding and detailed description of the interplay between the interference and nonlinearity in phonon propagation through and scattering in two-dimensional arrays of two-path anharmonic atomic defects with a different topology.

Ultradiscrete kinks with supersonic speed in a layered crystal with realistic potentials.

In this paper we develop a dynamical model of the propagating nonlinear localized excitations, supersonic kinks, in the cation layer in a silicate mica crystal. We start from purely electrostatic Coulomb interaction and add the Ziegler-Biersack-Littmark short-range repulsive potential and the periodic potential produced by other atoms of the lattice. The proposed approach allows the construction of supersonic kinks which can propagate in the lattice within a large range of energies and velocities. Due to the presence of the short-range repulsive component in the potential, the interparticle distances in the lattice kinks with high energy are limited by physically reasonable values. The introduction of the periodic lattice potential results in the important feature that the kinks propagate with the single velocity and single energy, which are independent on the excitation conditions. The unique average velocity of the supersonic kinks on the periodic substrate potential we relate with the kink amplitude of the relative particle displacements, which is determined by the interatomic distance corresponding to the minimum of the total, interparticle plus substrate, lattice potential. The found kinks are ultradiscrete and can be described with the "magic wave number" q=2π/3a, which was previously revealed in the nonlinear sinusoidal waves and supersonic kinks in the Fermi-Pasta-Ulam lattice. The extreme discreteness of the observed supersonic kinks, with basically two particles moving at the same time, allows the detailed interpretation of their double-kink structure, which is not possible for the multikinks without an account for the lattice discreteness. Analytical calculations of the displacement patterns and energies of the supersonic kinks are confirmed by numerical simulations. The computed energy of the found supersonic kinks in the considered realistic lattice potential is in a good agreement with the experimental evidence for the transport of localized energetic excitations in silicate mica crystals between the points of ^{40}K recoil and subsequent sputtering.

Surface acoustic BLOCH oscillations, the Wannier-Stark ladder, and Landau-Zener tunneling in a solid.

We present the experimental observation of Bloch oscillations, the Wannier-Stark ladder, and Landau-Zener tunneling of surface acoustic waves in perturbed grating structures on a solid substrate. A model providing a quantitative description of our experimental observations, including multiple Landau-Zener transitions of the anticrossed surface acoustic Wannier-Stark states, is developed. The use of a planar geometry for the realization of the Bloch oscillations and Landau-Zener tunneling allows a direct access to the elastic field distribution. The vertical surface displacement has been measured by interferometry.

Wandering breathers and self-trapping in weakly coupled nonlinear chains: classical counterpart of macroscopic tunneling quantum dynamics.

We present analytical and numerical studies of the phase-coherent dynamics of intrinsically localized excitations (breathers) in a system of two weakly coupled nonlinear oscillator chains. We show that there are two qualitatively different dynamical regimes of the coupled breathers, either immovable or slowly moving: the periodic transverse translation (wandering) of the low-amplitude breather between the chains and the one-chain-localization of the high-amplitude breather. These two modes of coupled nonlinear excitations, which involve a large number of anharmonic oscillators, can be mapped onto two solutions of a single pendulum equation, detached by a separatrix mode. We also show that these two regimes of coupled phase-coherent breathers are similar and are described by a similar pair of equations to the two regimes in the nonlinear tunneling dynamics of two weakly linked interacting (nonideal) Bose-Einstein condensates. On the basis of this profound analogy, we predict a tunneling mode of two weakly coupled Bose-Einstein condensates in which their relative phase oscillates around pi/2 mod pi. We also show that the magnitude of the static displacements of the coupled chains with nonlinear localized excitation, induced by the cubic term in the intrachain anharmonic potential, scales approximately as the total vibrational energy of the excitation, either a one- or two-chain one, and does not depend on the interchain coupling. This feature is also valid for a narrow stripe of several parallel-coupled nonlinear chains. We also study two-chain breathers which can be considered as bound states of discrete breathers, with different symmetry and center locations in the coupled chains, and bifurcation of the antiphase two-chain breather into the one-chain one. Bound states of two breathers with different commensurate frequencies are found in the two-chain system. Merging of two breathers with different frequencies into one breather in two coupled chains is observed. Wandering of the low-amplitude breather in a system of several, up to five, coupled nonlinear chains is studied, and the dependence of the wandering period on the number of chains is analytically estimated and compared with numerical results. The delocalizing transition of a one-dimensional (1D) breather in the 2D system of a large number of parallel-coupled nonlinear oscillator chains is described, in which the breather, initially excited in a given chain, abruptly spreads its vibrational energy in the whole 2D system upon decreasing the breather frequency or amplitude below the threshold one. The threshold breather frequency is above the cutoff phonon frequency in the 2D system, and the threshold breather amplitude scales as the square root of the interchain coupling constant. The delocalizing transition of the discrete vibrational breather in 2D and 3D systems of parallel-coupled nonlinear oscillator chains has an analogy with the delocalizing transition for Bose-Einstein condensates in 2D and 3D optical lattices.