Anderson localization in a disordered chain with a finite nonlinear response time

In this work, we investigate the competition of disorder, nonlinearity and non-adiabatic process on the wave packet dynamics in 1D. We follow the time evolution of the second moment of the wave packet distribution to characterize its spreading behavior. In order to describe the dynamical behavior of one-electron wave packets, we solve a discrete nonlinear Schr?dinger equation which effectively takes into account a diagonal disorder and a nonlinear contribution. Going beyond the adiabatic regime, we consider that the nonlinearity relaxes in time according to a Debye-like law. In the adiabatic regime, it has been recently demonstrated that the interplay of disorder and nonlinearity leads to a sub-diffusive spread of the wave packet. Here, we numerically demonstrate that no sub-diffusive spreading of the second moment of the wave packet distribution takes place when the finite response time of the nonlinearity is taken into account.     

Study of diffusion of wave packets in a square lattice under external fields along the discrete nonlinear Schrödinger equation

The object of the present work is to analyze the effect of nonlinearity on wave packet propagation in a square lattice subject to a magnetic and an electric field in the Hall configuration, by using the Discrete Nonlinear Schrödinger Equation (DNLSE). In previous works we have shown that without the nonlinear term, the presence of the magnetic field induces the formation of vortices that remain stationary, while a wave packet is introduced in the system. As for the effect of an applied electric field, it was shown that the vortices propagate in a direction perpendicular to the electric field, similar behavior as presented in the classical treatment, we provide a quantum mechanics explanation for that. We have performed the calculations considering first the action of the magnetic field as well as the nonlinearity. The results indicate that for low values of the nonlinear parameter U the vortices remain stationary while preserving the form. For greater values of the parameter the picture gets distorted, the more so, the greater the nonlinearity. As for the inclusion of the electric field, we note that for small U, the wave packet propagates perpendicular to the applied field, until for greater values of U the wave gets partially localized in a definite region of the lattice. That is, for strong nonlinearity the wave packet gets partially trapped, while the tail of it can propagate through the lattice. Note that this tail propagation is responsible for the over-diffusion for long times of the wave packet under the action of an electric field. We have produced short films that show clearly the time evolution of the wave packet, which can add to the understanding of the dynamics.     

Anderson localization or nonlinear waves: a matter of probability

In linear disordered systems Anderson localization makes any wave packet stay localized for all times. Its fate in nonlinear disordered systems (localization versus propagation) is under intense theoretical debate and experimental study. We resolve this dispute showing that, unlike in the common hypotheses, the answer is probabilistic rather than exclusive. At any small but finite nonlinearity (energy) value there is a finite probability for Anderson localization to break up and propagating nonlinear waves to take over. It increases with nonlinearity (energy) and reaches unity at a certain threshold, determined by the initial wave packet size. Moreover, the spreading probability stays finite also in the limit of infinite packet size at fixed total energy. These results generalize to higher dimensions as well.     

Pulse dynamics in an optical fiber with linear intermode coupling and Kerr nonlinearity dispersion

The effect of Kerr nonlinearity dispersion on the envelope duration and the velocity of the envelope maximum for a wave packet propagating in an optical fiber and formed by two coupled copropagating waves is studied. The critical (threshold) energy of wave packet collapse can substantially be diminished and the supraluminal mode of propagation of the pulse envelope maximum in a nonamplifying medium can be realized owing to a significantly higher nonlinearity dispersion in such systems in comparison with “single-wave” ones.     

Wave packet dynamics in a nonlinear tunnel-coupled structure of right- and left-handed media

The dynamics of a wave packet formed by tunnel-coupled forward and backward waves in a waveguide structure consisting of media with different signs of real parts of their refractive indices is investigated. Expressions for coupled-wave amplitude and reflection and transmission coefficients that are corrected for absorption are derived in the linear approximation. The expressions governing the wave-packet duration and propagation velocity of its envelope maximum are derived, taking into account the second-order dispersion, cubic nonlinearity, and dispersion of the nonlinearity. The possibility of efficient control of the forward wave velocity by applying an external magnetic field is demonstrated.     

Electron wave packet dynamics in twisted nonlinear ladders with correlated disorder

Within a tight-binding Hamiltonian approach, we study the dynamics of one-electron wave packets in a twisted ladder geometry with adiabatic electron-phonon interaction. The electron-phonon coupling is taken into account in the time-dependent Schrödinger equation through a cubic nonlinearity. This physical scenario incorporates several relevant ingredients to study the electronic wave packet dynamics in DNA-like segments. In the absence of nonlinearity, a random sequence of nucleotides pairs makes the wave packets remain localized, according to the standard picture of the Anderson localization. However, when the electron-phonon interaction is turned on, Anderson localization is suppressed and a subdiffusive regime takes place. Further, we show that the wave packet trapping can be controlled by an external field perpendicular to the helicity axis of the double-strand chain.     

Dynamics of optical pulses in periodic nonlinear fibers

Dynamics of a two-mode wave packet with strong linear and nonlinear (cross-modulation) coupling is investigated. The effect of the initial conditions of the pulse excitation on its further transformation is considered. The possibility to control the dispersion parameters of the wave packet by varying the initial conditions of its input is pointed out. The effective parameters of the dispersion and nonlinearity that govern the dynamics of an optical pulse in a periodic nonlinear fiber light guide are obtained.     

Self-focusing of wave packets and envelope shock formation in nonlinear dispersive media

The self-action of three-dimensional wave packets is analyzed analytically and numerically under the conditions of competing diffraction, cubic nonlinearity, and nonlinear dispersion (dependence of group velocity on wave amplitude). A qualitative analysis of pulse evolution is performed by the moment method to find a sufficient condition for self-focusing. Self-action effects in an electromagnetically induced transparency medium (without cubic nonlinearity) are analyzed numerically. It is shown that the self-focusing of a wave packet is accompanied by self-steepening of the longitudinal profile and envelope shock formation. The possibility of envelope shock formation is also demonstrated for self-focusing wave packets propagating in a normally dispersive medium.     

Absence of wave packet diffusion in disordered nonlinear systems

We study the spreading of an initially localized wave packet in two nonlinear chains (discrete nonlinear Schr?dinger and quartic Klein-Gordon) with disorder. Previous studies suggest that there are many initial conditions such that the second moment of the norm and energy density distributions diverges with time. We find that the participation number of a wave packet does not diverge simultaneously. We prove this result analytically for norm-conserving models and strong enough nonlinearity. After long times we find a distribution of nondecaying yet interacting normal modes. The Fourier spectrum shows quasiperiodic dynamics. Assuming this result holds for any initially localized wave packet, we rule out the possibility of slow energy diffusion. The dynamical state could approach a quasiperiodic solution (Kolmogorov-Arnold-Moser torus) in the long time limit.     

High-Dimensional Nonlinear Envelope Equations and Nonlinear Localized Excitations in Photonic Crystals

We investigate the nonlinear localized structures of optical pulses propagating in a one-dimensional photonic crystal with a quadratic nonlinearity. Using a method of multiple scales we show that the nonlinear evolution of a wave packet, formed by the superposition of short-wavelength excitations, and long-wavelength mean fields, generated by the self-interaction of the wave packet, are governed by a set of coupled high-dimensional nonlinear envelope equations, which can be reduced to Davey-Stewartson equations and thus support dromionlike high-dimensional nonlinear excitations in the system.     

High-Dimensional Nonlinear Envelope Equations and Nonlinear Localized Excitations in Photonic Crystals

We investigate the nonlinear localized structures of optical pulses propagating in a one-dimensional photonic crystal with a quadratic nonlinearity. Using a method of multiple scales we show that the nonlinear evolution of a wave packet, formed by the superposition of short-wavelength excitations, and long-wavelength mean fields, generated by the self-interaction of the wave packet, are governed by a set of coupled high-dimenslonal nonlinear envelope equations, which can be reduced to Davey-Stewartson equations and thus support dromionlike high-dimensional nonlinear excitations in the system.     

Dynamics of two-mode radiation in optical waveguides with strong mode coupling

Amplitude equations, as well as the effective dispersion and nonlinearity parameters, which define the dynamics of a wave packet formed by two strongly coupled modes, are derived with allowance for the frequency dependence of the linear mode coupling coefficient. These equations are used to study the onset of the modulation instability of the two-mode wave packet, soliton-like pulses, and compression modes. Unlike single-mode systems, the last two effects in optical waveguides may arise for both a negative and positive disper-sion of the waveguide material.     

Random generation of coherent solitary waves from incoherent waves

The random generation of coherent solitary waves from incoherent waves in a medium with an instantaneous nonlinearity has been observed. One excites a propagating incoherent spin wave packet in a magnetic film strip and observes the random appearance of solitary wave pulses. These pulses are as coherent as traditional solitary waves, but with random timing and a random peak amplitude.     

Radiation of caustic beams from a collapsing bullet

We show both theoretically and experimentally that a collapsing (2+1)-dimensional wave packet in a medium with cubic nonlinearity and a two-dimensional dispersion of an order higher than parabolic irradiates untrapped dispersive waves. The studies are performed for a spin-wave bullet propagating in an in-plane magnetized ferrimagnetic film. An induced uniaxial anisotropy in such a medium leads to the formation of narrow spin-wave caustic beams whose angles to the bullet's propagation direction are modified by the motion of the source.     

Two kinds of upper hybrid soliton bistability

The basic model employed to describe nonlinear upper hybrid wave structures is the generalized nonlinear Schrödinger equation including second and fourth order dispersive effects as well as local and nonlocal nonlinearity. For two kinds of such an equation the existence of two stable solitons with the same plasmon number but with different spatial scales and amplitudes is shown as two qualitatively different kinds of upper hybrid soliton bistability. An integral relation for an arbitrary nonlinear upper hybrid wave packet evolution is derived taking into account higher order dispersive effects. Necessary conditions for soliton formation from arbitrary wave packets and the impossibility of wave packet collapse are demonstrated taking into account higher order dispersive effects.     

Destruction of Anderson localization by a weak nonlinearity

We study numerically the spreading of an initially localized wave packet in a one-dimensional discrete nonlinear Schr?dinger lattice with disorder. We demonstrate that above a certain critical strength of nonlinearity the Anderson localization is destroyed and an unlimited subdiffusive spreading of the field along the lattice occurs. The second moment grows with time proportional, variant t alpha, with the exponent alpha being in the range 0.3-0.4. For small nonlinearities the distribution remains localized in a way similar to the linear case.     

Nonlinear delocalization on disordered Stark ladder

We study effects of weak nonlinearity on localization of waves in disordered Stark ladder corresponding to propagation in presence of disorder and a static field. Our numerical results show that nonlinearity leads to delocalization with subdiffusive spreading along the ladder. The exponent of spreading remains close to its value in absence of the static field. The delocalization implies the existence of statistical entanglement between far away parts of the spreading wave packet indicating importance of long-range effects.     

Horizontal rolls in convective flow above a partially heated surface

Considering the nonlinearity arising from the interaction between electrons and lattice vibrations, an effective electronic model with a self-interaction cubic term is employed to study the interplay between electron-electron and electron-phonon interactions. Based on numerical solutions of the time-dependent nonlinear Schroedinger equation for an initially localized two-electron singlet state, we show that the magnitude of the electron-phonon coupling χ necessary to promote the self-trapping of the electronic wave packet decreases as a function of the electron-electron interaction U. We show that such dependence is directly linked to the narrowing of the band of bounded two-electron states as U increases. We obtain the transition line in the χ × U parameter space separating the phases of self-trapped and delocalized electronic wave packets. The present results indicates that nonlinear contributions plays a relevant role in the electronic wave packet dynamics, particularly in the regime of strongly correlated electrons.     

On the collapse of wave packets in a medium with normal group velocity dispersion

A solution to the problem of realizing the collapse of three-dimensional wave packets in nonlinear media with normal group velocity dispersion is proposed. Wave packets with pronounced hyperbolic topology are shown to collapse; i.e., the field increases infinitely near the system axis. In particular, wave collapse of the tubular axisymmetric packets occurs through the concentration of the compressed ring field distribution at the axis. The collapse is shown to stabilize due to the saturation of nonlinearity or nonlinear dissipation, which restrict the field increase and lead to the packet splitting in the transverse direction.     

Coherent forward scattering of a single photon wave packet in a resonant medium

The propagation of a single photon wave packet in a thick resonant medium was studied theoretically and experimentally. The well-known acceleration of wave packet decay was observed upon wave packet exiting from a medium with single resonant absorption line. It was found that in a medium with two closely spaced absorption lines, the restoration of the photon wave packet is observed after the acceleration of decay.