Nonlinearity-induced localization in a periodically driven semidiscrete system.

We demonstrate that nonlinearity plays a constructive role in supporting the robustness of dynamical localization in a system which is discrete in one dimension and continuous in the orthogonal one. In the linear regime, time-periodic modulation of the gradient strength along the discrete axis leads to the usual rapid spread of an initially confined wave packet. Addition of the cubic nonlinearity makes the dynamics drastically different, inducing robust localization of moving wave packets. Similar nonlinearity-induced effects are also produced in the presence of a combination of static and oscillating linear potentials. The predicted dynamical localization in the nonlinear medium can be realized in photonic lattices and Bose-Einstein condensates.

Bloch oscillations sustained by nonlinearity.

We demonstrate that nonlinearity may play a constructive role in supporting Bloch oscillations in a model which is discrete, in one dimension and continuous in the orthogonal one. The model can be experimentally realized in several fields of physics such as optics and Bose-Einstein condensates. We demonstrate that designing an optimal relation between the nonlinearity and the linear gradient strength provides extremely long-lived Bloch oscillations with little degradation. Such robust oscillations can be observed for a broad range of parameters and even for moderate nonlinearities and large enough values of linear potential. We also present an approximate analytical description of the wave packet's evolution featuring a hybrid Bloch oscillating wave-soliton behavior that excellently corresponds to the direct numerical simulations.

Dynamics of dipoles and vortices in nonlinearly coupled three-dimensional field oscillators.

The dynamics of a pair of harmonic oscillators represented by three-dimensional fields coupled with a repulsive cubic nonlinearity is investigated through direct simulations of the respective field equations and with the help of the finite-mode Galerkin approximation (GA), which represents the two interacting fields by a superposition of 3+3 harmonic-oscillator p-wave eigenfunctions with orbital and magnetic quantum numbers l=1 and m=1, 0, -1. The system can be implemented in binary Bose-Einstein condensates, demonstrating the potential of the atomic condensates to emulate various complex modes predicted by classical field theories. First, the GA very accurately predicts a broadly degenerate set of the system's ground states in the p-wave manifold, in the form of complexes built of a dipole coaxial with another dipole or vortex, as well as complexes built of mutually orthogonal dipoles. Next, pairs of noncoaxial vortices and/or dipoles, including pairs of mutually perpendicular vortices, develop remarkably stable dynamical regimes, which feature periodic exchange of the angular momentum and periodic switching between dipoles and vortices. For a moderately strong nonlinearity, simulations of the coupled-field equations agree very well with results produced by the GA, demonstrating that the dynamics is accurately spanned by the set of six modes limited to l=1.

Emulation of Fabry-Perot and Bragg resonators with temporal optical solitons.

The scattering of weak dispersive waves (DWs) on several equally spaced temporal solitons is studied. It is shown by systematic numerical simulations that the reflection of the DWs from the soliton trains strongly depends on the distance between the solitons. The dependence of the reflection and transmission coefficients on the inter-soliton distance and the frequency of the incident waves are studied in detail, revealing fascinating quasi-periodic behavior. The analogy between the observed nonlinear phenomena in the temporal domain and the usual Fabry-Perot and Bragg resonators is discussed.

Precession and nutation dynamics of nonlinearly coupled non-coaxial three-dimensional matter wave vortices.

Nonlinearity is the driving force for numerous important effects in nature typically showing transitions between different regimes, regular, chaotic or catastrophic behavior. Localized nonlinear modes have been the focus of intense research in areas such as fluid and gas dynamics, photonics, atomic and solid state physics etc. Due to the richness of the behavior of nonlinear systems and due to the severe numerical demands of accurate three-dimensional (3D) numerical simulations presently only little knowledge is available on the dynamics of complex nonlinear modes in 3D. Here, we investigate the dynamics of 3D non-coaxial matter wave vortices that are trapped in a parabolic potential and interact via a repulsive nonlinearity. Our numerical simulations demonstrate the existence of an unexpected and fascinating nonlinear regime that starts immediately when the nonlinearity is switched-on and is characterized by a smooth dynamics representing torque-free precession with nutations. The reported motion is proven to be robust regarding various effects such as the number of particles, dissipation and trap deformations and thus should be observable in suitably designed experiments. Since our theoretical approach, i.e., coupled nonlinear Schrödinger equations, is quite generic, we expect that the obtained novel dynamical behavior should also exist in other nonlinear systems.

Interaction of high-order solitons with external dispersive waves.

The effect of mutual interaction between second-order soliton and dispersive waves (DWs) is investigated. It is predicted analytically and confirmed numerically that DWs (both transmitted and reflected components) become polychromatic after interaction with the soliton. Collision with DWs of considerable intensity can lead to acceleration/deceleration and central frequency shift of the soliton, while still preserving the soliton's oscillating structure. Two second-order solitons with resonant DWs trapped between them can form an effective solitonic cavity with "flat" or "concave mirrors," depending on the intensity of the input.

Weak and strong interactions between dark solitons and dispersive waves.

The effect of mutual interactions between dark solitons and dispersive waves is investigated numerically and analytically. The condition of the resonant scattering of dispersive waves on dark solitons is derived and compared against the results of the numerical simulations. It is shown that the interaction with intense dispersive waves affects the dynamics of the solitons by accelerating, decelerating, or destroying them. It is also demonstrated that two dark solitons can form a cavity for dispersive waves bouncing between the two dark solitons. The differences of the resonant scattering of the dispersive waves on dark and bright solitons are discussed. In particular, we demonstrate that two dark solitons and a dispersive wave bouncing in between them create a solitonic cavity with convex "mirrors," unlike the concave "mirror" in the case of bright solitons.

Resonant radiation from oscillating higher order solitons.

We present radiation mechanism exhibited by a higher order soliton. In a course of its evolution the higher-order soliton emits polychromatic radiation resulting in formation of multipeak frequency comb-like spectral band. The shape and spectral position of this band can be effectively controlled by the relative strength of the third order dispersion. An analytical description is corroborated by numerical simulations. It is shown that for longer pulses the described effect persists also under the action of higher order perturbations such as Raman and self-steepening.

Coupled Airy breathers.

The dynamics of two component-coupled vectorial Airy beams is investigated. In the linear propagation regime, a complete analytic solution describes the breather-like propagation of two components that feature nondiffracting self-accelerating Airy behavior. The superposition of two beams with different input properties opens the possibility of designing more complex nondiffracting propagation scenarios. In the strongly nonlinear regime, the dynamics remain qualitatively robust as is revealed by direct numerical simulations. Because of the Kerr effect, the two beams emit solitonic breathers whose coupling period is compatible with the remaining Airy-like beams. The results of this study are relevant for the description of photonic and plasmonic beams that propagate in coupled planar waveguides, as well as for birefrigent or multiwavelength beams.

Tuning resonant interaction of orthogonally polarized solitons and dispersive waves with the soliton power.

We demonstrate that the relatively small power induced changes in the soliton wavenumber comparable with splitting of the effective indexes of the orthogonally polarized waveguide modes result in significant changes of the efficiency of the interaction between solitons and dispersive waves and can be used to control energy transfer between the soliton and newly generated waves and to delay or accelerate solitons.

Trapping of light in solitonic cavities and its role in the supercontinuum generation.

We demonstrate that the fission of higher-order N-solitons with a subsequent ejection of fundamental quasi-solitons creates cavities formed by a pair of solitary waves with dispersive light trapped between them. As a result of multiple reflections of the trapped light from the bounding solitons which act as mirrors, they bend their trajectories and collide. In the spectral domain, the two solitons receive blue and red wavelength shifts, and the spectrum of the trapped light alters as well. This phenomenon strongly affects spectral characteristics of the generated supercontinuum. Consideration of the system's parameters which affect the creation of the cavity reveals possibilities of predicting and controlling soliton-soliton collisions induced by multiple reflections of the trapped light.

Soliton interaction mediated by cascaded four wave mixing with dispersive waves.

We demonstrate that trapping of dispersive waves between two optical solitons takes place when resonant scattering of the waves on the solitons leads to nearly perfect reflections. The momentum transfer from the radiation to solitons results in their mutual attraction and a subsequent collision. The spectrum of the trapped radiation can either expand or shrink in the course of the propagation, which is controlled by arranging either collision or separation of the solitons.

Solitary pulse propagation and soliton-induced supercontinuum generation in silica glasses containing silver nanoparticles.

We study solitary pulse propagation in the normal dispersion region in glasses containing silver nanoparticles with a self-defocusing nonlinearity and predict that, despite high plasmonic loss, pulse propagation without significant distortion over five soliton periods can be achieved in such materials. As an application, we study low-threshold soliton-induced supercontinuum generation and predict more than octave-spanning spectral broadening by femtosecond pulses with an intensity in the range of hundreds of GW/cm(2).

Supercontinuum generation in aqueous colloids containing silver nanoparticles.

We numerically study low-threshold supercontinuum generation using the significant enhancement of nonlinearity in aqueous colloids with silver nanoparticles. We predict octave-spanning spectral broadening by femtosecond pulses with an intensity in the range of tens of GW/cm2. The strong frequency dependence of the effective nonlinear coefficient of the composite significantly influences the spectral broadening by self-phase modulation and leads to a large blueshift of the spectra.

Mismatch management for optical and matter-wave quadratic solitons.

We propose a way to control solitons in chi(2) (quadratically nonlinear) systems by means of periodic modulation imposed on the phase-mismatch parameter ("mismatch management," MM). It may be realized in the cotransmission of fundamental-frequency (FF) and second-harmonic (SH) waves in a planar optical waveguide via a long-period modulation of the usual quasi-phase-matching pattern of ferroelectric domains. In an altogether different physical setting, the MM may also be implemented by dint of the Feshbach resonance in a harmonically modulated magnetic field in a hybrid atomic-molecular Bose-Einstein condensate (BEC), with the atomic and molecular mean fields (MFs) playing the roles of the FF and SH, respectively. Accordingly, the problem is analyzed in two different ways. First, in the optical model, we identify stability regions for spatial solitons in the MM system, in terms of the MM amplitude and period, using the MF equations for spatially inhomogeneous configurations. In particular, an instability enclave is found inside the stability area. The robustness of the solitons is also tested against variation of the shape of the input pulse, and a threshold for the formation of stable solitons is found in terms of the power. Interactions between stable solitons are virtually unaffected by the MM. The second method (parametric approximation), going beyond the MF description, is developed for spatially homogeneous states in the BEC model. It demonstrates that the MF description is valid for large modulation periods, while, at smaller periods, non-MF components acquire gain, which implies destruction of the MF under the action of the high-frequency MM.