Nonlinear dynamics of femtosecond optical solitary wave propagationat the zero dispersion point

A perturbed nonlinear Schrodinger equation, describing nonlinear pulse propagation of femtosecond duration in optical fibers at the zero dispersion point, is considered. This equation is reduced to an ordinary differential equation, which is treated by means of Hamiltonian dynamics techniques. Conditions for solitary wave formation, as well as the solutions themselves, namely, a bright soliton, a rarefaction soliton, and a pair of dark solitons, are derived analytically. The shifts of the solitary wave velocity, the angular frequency, and the wavenumber, as well as the peak power required to launch a bright soliton are also obtained. The connection among the point initial conditions of the pulse and the type of solitary wave that can be excited is finally presented     

Necklacelike solitons in optically induced photonic lattices

We report the first observation of stationary necklacelike solitons. Such necklace structures were realized when a high-order vortex beam was launched appropriately into a two-dimensional optically induced photonic lattice. Our theoretical results obtained with continuous and discrete models show that the necklace solitons resulting from a charge-4 vortex have a pi phase difference between adjacent "pearls" and are formed in an octagon shape. Their stability region is identified.     

Three-dimensional solitary waves and vortices in a discrete nonlinear Schrödinger lattice

In a benchmark dynamical-lattice model in three dimensions, the discrete nonlinear Schr?dinger equation, we find discrete vortex solitons with various values of the topological charge S. Stability regions for the vortices with S=0,1,3 are investigated. The S=2 vortex is unstable and may spontaneously rearranging into a stable one with S=3. In a two-component extension of the model, we find a novel class of stable structures, consisting of vortices in the different components, perpendicularly oriented to each other. Self-localized states of the proposed types can be observed experimentally in Bose-Einstein condensates trapped in optical lattices and in photonic crystals built of microresonators.     

Feshbach resonance management for Bose-Einstein condensates

An experimentally realizable scheme of periodic sign-changing modulation of the scattering length is proposed for Bose-Einstein condensates similar to dispersion-management schemes in fiber optics. Because of controlling the scattering length via the Feshbach resonance, the scheme is named Feshbach-resonance management. The modulational-instability analysis of the quasiuniform condensate driven by this scheme leads to an analog of the Kronig-Penney model. The ensuing stable localized structures are found. These include breathers, which oscillate between the Thomas-Fermi and Gaussian configuration, or may be similar to the 2-soliton state of the nonlinear Schr?dinger equation, and a nearly static state ("odd soliton") with a nested dark soliton. An overall phase diagram for breathers is constructed, and full stability of the odd solitons is numerically established.     

Dark solitons in mode-locked lasers

Dark soliton formation in mode-locked lasers is investigated by means of a power-energy saturation model that incorporates gain and filtering saturated with energy, and loss saturated with power. It is found that general initial conditions evolve (mode-lock) into dark solitons under appropriate requirements also met in experimental observations. The resulting pulses are essentially dark solitons of the unperturbed nonlinear Schr?dinger equation. Notably, the same framework also describes bright pulses in anomalous and normally dispersive lasers.     

Effects of long-range nonlinear interactions in double-well potentials

We consider the interplay of linear double-well-potential (DWP) structures and nonlinear long-range interactions of different types, motivated by applications to nonlinear optics and matter waves. We find that, while the basic spontaneous-symmetry-breaking (SSB) bifurcation structure in the DWP persists in the presence of the long-range interactions, the critical points at which the SSB emerges are sensitive to the range of the nonlocal interaction. We quantify the dynamics by developing a few-mode approximation corresponding to the DWP structure, and analyze the resulting system of ordinary differential equations and its bifurcations in detail. We compare results of this analysis with those produced by the full partial differential equation, finding good agreement between the two approaches. Effects of the competition between the local self-attraction and nonlocal repulsion on the SSB are studied too. A far more complex bifurcation structure involving the possibility for not only supercritical but also subcritical bifurcations and even bifurcation loops is identified in that case.     

Backward-wave propagation and discrete solitons in a left-handed electrical lattice

We study experimentally, analytically and numerically the backward-wave propagation, and formation of discrete bright and dark solitons in a nonlinear electrical lattice. We observe experimentally that a focusing (defocusing) effect occurs above (below) a certain carrier frequency threshold, and backward-propagating bright (dark) discrete solitons are formed. We develop a discrete model emulating the relevant circuit and benchmark its linear properties against the experimental dispersion relation. Using a perturbation method, we derive a nonlinear Schrödinger equation, that predicts accurately the carrier frequency threshold. Finally, we use numerical simulations to corroborate our findings and monitor the space-time evolution of the discrete solitons.     

Dynamics of dark-bright solitons in cigar-shaped Bose-Einstein condensates

We explore the stability and dynamics of dark-bright (DB) solitons in two-component elongated Bose-Einstein condensates by developing effective one-dimensional vector equations and solving the three-dimensional Gross-Pitaevskii equations. A strong dependence of the oscillation frequency and of the stability of the DB soliton on the atom number of its components is found; importantly, the wave may become dynamically unstable even in the 1D regime. As the atom number in the dark-soliton-supporting component is further increased, spontaneous symmetry breaking leads to oscillatory dynamics in the transverse degrees of freedom. Moreover, the interactions of two DB solitons are investigated with an emphasis on the importance of their relative phases. Experimental results showcasing multiple DB soliton oscillations and a DB-DB collision in a Bose-Einstein condensate consisting of two hyperfine states of ⁸⁷Rb confined in an elongated optical dipole trap are presented.     

On the formation of gap solitons in nonlinear electromagnetic metamaterials

A nonlinear (Kerr‐type) electromagnetic metamaterial, characterized by a double‐Lorentz model of its frequency‐dependent linear effective dielectric permittivity and magnetic permeability, is considered. The formation of gap solitons in the low‐ and high‐frequency band gaps of this metamaterial is investigated analytically. Evolution equations governing the gap solitons, of the form of a nonlinear Klein‐Gordon and a nonlinear Schrödinger equation, are obtained, and the structure of their solutions is discussed.     

Disk-shaped Bose-Einstein condensates in the presence of an harmonic trap and an optical lattice

We study the existence and stability of solutions of the two-dimensional nonlinear Schrodinger equation in the combined presence of a parabolic and a periodic potential. The motivating physical example consists of Bose-Einstein condensates confined in an harmonic (e.g., magnetic) trap and an optical lattice. By connecting the nonlinear problem with the underlying linear spectrum, we examine the bifurcation of nonlinear modes out of the linear ones for both focusing and defocusing nonlinearities. In particular, we find real-valued solutions (such as multipoles) and complex-valued ones (such as vortices). A primary motivation of the present work is to develop "rules of thumb" about what waveforms to expect emerging in the nonlinear problem and about the stability of those modes. As a case example of the latter, we find that among the real-valued solutions, the one with larger norm for a fixed value of the chemical potential is expected to be unstable.