Ramaz Khomeriki

Nonlinear supratransmission and bistability in the Fermi-Pasta-Ulam model

The recently discovered phenomenon of nonlinear supratransmission consists in a sudden increase of the amplitude of a transmitted wave triggered by the excitation of nonlinear localized modes of the medium. We examine this process for the Fermi-Pasta-Ulam chain, sinusoidally driven at one edge and damped at the other. The supratransmission regime occurs for driving frequencies above the upper band edge and originates from direct moving discrete breather creation. We derive approximate analytical estimates of the supratransmission threshold, which are in excellent agreement with numerics. When analyzing the long-time behavior, we discover that, below the supratransmission threshold, a conducting stationary state coexists with the insulating one. We explain the bistable nature of the energy flux in terms of the excitation of quasiharmonic extended waves. This leads to the analytical calculation of a lower-transmission threshold which is also in reasonable agreement with numerical experiments.

The anti-FPU problem

We present a detailed analysis of the modulational instability of the zone-boundary mode for one and higher-dimensional Fermi-Pasta-Ulam (FPU) lattices. Following this instability, a process of relaxation to equipartition takes place, which we have called the Anti-FPU problem because the energy is initially fed into the highest frequency part of the spectrum, at variance with the original FPU problem (low frequency excitations of the lattice). This process leads to the formation of chaotic breathers in both one and two dimensions. Finally, the system relaxes to energy equipartition on time scales which increase as the energy density is decreased. We show that breathers formed when cooling the lattice at the edges, starting from a random initial state, bear strong qualitative similarities with chaotic breathers.

Nonlinear band gap transmission in optical waveguide arrays.

The effect of nonlinear transmission in coupled optical waveguide arrays is theoretically investigated and a realistic experimental setup is suggested. The beam is injected in a single boundary waveguide, linear refractive index of which (n(0)) is larger than refractive indexes (n) of other identical waveguides in the array. Particularly, the effect holds if omega(n(0)-n)/c>2Q, where Q is a linear coupling constant between array waveguides, omega is a carrier wave frequency, and c is a light velocity. Numerical experiments show that the energy transfers from the boundary waveguide to the waveguide array above a certain threshold intensity of the injected beam. This effect is due to the creation and the propagation of gap solitons in full analogy with a similar phenomenon in sine-Gordon lattice [Phys. Rev. Lett. 89, 134102 (2002)]].

Bistability in the sine-gordon equation : The ideal switch

The sine-Gordon equation, used as the representative nonlinear wave equation, presents a bistable behavior resulting from nonlinearity and generating hysteresis properties. We show that the process can be understood in a comprehensive analytical formulation and that it is a generic property of nonlinear systems possessing a natural band gap. The approach allows one to discover that the sine-Gordon equation can work as an ideal switch by reaching a transmissive regime with vanishing driving amplitude.

Correlated metallic two-particle bound states in quasiperiodic chains

Single-particle states in a chain with quasiperiodic potential show a metal-insulator transition upon the change of the potential strength. We consider two particles with local interaction in the single-particle insulating regime. The two-particle states change from being localized to delocalized upon an increase of the interaction strength to a nonperturbative finite value. At even larger interaction strength the states become localized again. This transition of two-particle bound states into correlated metallic ones is due to a resonant mixing of the noninteracting two-particle eigenstates. In the discovered correlated metal states two particles move coherently together through the whole chain.

Delocalization and spreading in a nonlinear Stark ladder

We study the evolution of a wave packet in a nonlinear Stark ladder. In the absence of nonlinearity all normal modes are spatially localized giving rise to an equidistant eigenvalue spectrum and Bloch oscillations. Nonlinearity induces frequency shifts and mode-mode interactions and destroys localization. For large strength of nonlinearity we observe single-site trapping as a transient, with subsequent explosive spreading, followed by subdiffusion. For moderate nonlinearities an immediate subdiffusion takes place. Finally, for small nonlinearities we find linear Stark localization as a transient, with subsequent subdiffusion. For single-mode excitations and weak nonlinearities, stability intervals are predicted and observed upon variation in the dc bias strength, which affects the short- and the long-time dynamics.

Landau-Zener Bloch Oscillations with Perturbed Flat Bands.

Sinusoidal Bloch oscillations appear in band structures exposed to external fields. Landau-Zener (LZ) tunneling between different bands is usually a counteracting effect limiting Bloch oscillations. Here we consider a flat band network with two dispersive and one flat band, e.g., for ultracold atoms and optical waveguide networks. Using external synthetic gauge and gravitational fields we obtain a perturbed yet gapless band structure with almost flat parts. The resulting Bloch oscillations consist of two parts-a fast scan through the nonflat part of the dispersion structure, and an almost complete halt for substantial time when the atomic or photonic wave packet is trapped in the original flat band part of the unperturbed spectrum, made possible due to LZ tunneling.

Dynamics of localized modes in a composite multiferroic chain.

In a coupled ferroelectric-ferromagnetic system, i.e., a composite multiferroic, the propagation of magnetic or ferroelectric excitations across the whole structure is a key issue for applications. Of special interest is the dynamics of localized magnetic or ferroelectric modes (LM) across the ferroelectric-ferromagnetic interface, particularly when the LMs carrier frequency is in the band of the ferroelectric and in the band gap of the ferromagnet. For a proper choice of the systems parameters, we find that there is a threshold amplitude above which the interface becomes transparent and an in-band ferroelectric LM penetrates the ferromagnetic array. Below that threshold, the LM is fully reflected. Slightly below this transmission threshold, the addition of noise may lead to energy transmission, provided that the noise level is neither too low nor too high, an effect that resembles stochastic resonance. These findings represent an important step towards the application of ferroelectric and/or ferromagnetic LM-based logic.

Light Detectors Bistable Nonlinear Waveguide Arrays

Bistability induced by nonlinear Kerr effect in arrays of coupled waveguides is studied and shown to be a means to conceive light detectors that switch under excitation by a weak signal. The detector is obtained by coupling two single 1D waveguide to an array of coupled waveguides with adjusted indices and coupling. The process is understood by analytical description in the conservative and continuous case and illustrated by numerical simulations of the model with attenuation.

Traveling solitons in long-range oscillator chains

We investigate the existence and propagation of solitons in a long-range extension of the quartic Fermi-Pasta-Ulam (FPU) chain of anharmonic oscillators. The coupling in the linear term decays as a power-law with an exponent greater than 1 and less than 3. We obtain an analytic perturbative expression of traveling envelope solitons by introducing a Non Linear Schrodinger (NLS) equation for the slowly varying amplitude of short wavelength modes. Due to the non analytic properties of the dispersion relation, it is crucial to develop the theory using discrete difference operators. Those properties are also the ultimate reason why kink-solitons may exist but are unstable, at variance with the short-range FPU model. We successfully compare these approximate analytic results with numerical simulations.