Solitons supported by localized nonlinearities in periodic media

Nonlinear periodic systems, such as photonic crystals and Bose-Einstein condensates (BECs) loaded into optical lattices, are often described by the nonlinear Schr\"odinger/Gross-Pitaevskii equation with a sinusoidal potential. Here, we consider a model based on such a periodic potential, with the nonlinearity (attractive or repulsive) concentrated either at a single point or at a symmetric set of two points, which are represented, respectively, by a single {\delta}-function or a combination of two {\delta}-functions. This model gives rise to ordinary solitons or gap solitons (GSs), which reside, respectively, in the semi-infinite or finite gaps of the system's linear spectrum, being pinned to the {\delta}-functions. Physical realizations of these systems are possible in optics and BEC, using diverse variants of the nonlinearity management. First, we demonstrate that the single {\delta}-function multiplying the nonlinear term supports families of stable regular solitons in the self-attractive case, while a family of solitons supported by the attractive {\delta}-function in the absence of the periodic potential is completely unstable. We also show that the {\delta}-function can support stable GSs in the first finite gap in both the self-attractive and repulsive models. The stability analysis for the GSs in the second finite gap is reported too, for both signs of the nonlinearity. Alongside the numerical analysis, analytical approximations are developed for the solitons in the semi-infinite and first two finite gaps, with the single {\delta}-function positioned at a minimum or maximum of the periodic potential. In the model with the symmetric set of two {\delta}-functions, we study the effect of the spontaneous symmetry breaking of the pinned solitons. Two configurations are considered, with the {\delta}-functions set symmetrically with respect to the minimum or maximum of the potential

A nonpolynomial Schroedinger equation for resonantly absorbing gratings

We derive a nonlinear Schroedinger equation with a radical term, in the form of the square root of (1-|V|^2), as an asymptotic model of the optical medium built as a periodic set of thin layers of two-level atoms, resonantly interacting with the electromagnetic field and inducing the Bragg reflection. A family of bright solitons is found, which splits into stable and unstable parts, exactly obeying the Vakhitov-Kolokolov criterion. The soliton with the largest amplitude, which is |V| = 1, is found in an explicit analytical form. It is a "quasi-peakon", with a discontinuity of the third derivative at the center. Families of exact cnoidal waves, built as periodic chains of quasi-peakons, are found too. The ultimate solution belonging to the family of dark solitons, with the background level |V| = 1, is a dark compacton, also obtained in an explicit analytical form. Those bright solitons which are unstable destroy themselves (if perturbed) attaining the critical amplitude, |V| = 1. The dynamics of the wave field around this critical point is studied analytically, revealing a switch of the system into an unstable phase. Collisions between bright solitons are investigated too. The collisions between fast solitons are quasi-elastic, while slowly moving ones merge into breathers, which may persist or perish (in the latter case, also by attaining |V| = 1).Comment: Physical Review A, in pres

Self-induced transparency and giant nonlinearity in doped photonic crystals

Photonic crystals doped with resonant atoms allow for uniquely advantageous nonlinear modes of optical propagation: (a) Self-induced transparency (SIT) solitons and multi-dimensional localized "bullets" propagating at photonic band gap frequencies. These modes can exist even at ultraweak intensities (few photons) and therefore differ substantially either from solitons in Kerr-nonlinear photonic crystals or from SIT solitons in uniform media. (b) Cross-coupling between pulses exhibiting electromagnetically induced transparency (EIT) and SIT gap solitons. We show that extremely strong correlations (giant cross-phase modulation) can be formed between the two pulses. These features may find applications in high-fidelity classical and quantum optical communications.Comment: 11 pages, 7 figures, to appear in JOSA-

Scattering of slow-light gap solitons with charges in a two-level medium

The Maxwell-Bloch system describes a quantum two-level medium interacting with a classical electromagnetic field by mediation of the the population density. This population density variation is a purely quantum effect which is actually at the very origin of nonlinearity. The resulting nonlinear coupling possesses particularly interesting consequences at the resonance (when the frequency of the excitation is close to the transition frequency of the two-level medium) as e.g. slow-light gap solitons that result from the nonlinear instability of the evanescent wave at the boundary. As nonlinearity couples the different polarizations of the electromagnetic field, the slow-light gap soliton is shown to experience effective scattering whith charges in the medium, allowing it for instance to be trapped or reflected. This scattering process is understood qualitatively as being governed by a nonlinear Schroedinger model in an external potential related to the charges (the electrostatic permanent background component of the field).Comment: RevTex, 14 pages with 5 figures, to appear in J. Phys. A: Math. Theo

Spatiotemporally localized solitons in resonantly absorbing Bragg reflectors

We predict the existence of spatiotemporal solitons (``light bullets'') in two-dimensional self-induced transparency media embedded in a Bragg grating. The "bullets" are found in an approximate analytical form, their stability being confirmed by direct simulations. These findings suggest new possibilities for signal transmission control and self-trapping of light.Comment: RevTex, 3 pages, 2 figures, to be published in PR

Control of Azomethine Cycloaddition Stereochemistry by CF<inf>3</inf> Group: Structural Diversity of Fluorinated β-Proline Dimers

© 2016 American Chemical Society.β-Proline-functionalized dimers consisting of homochiral monomeric units were synthesized by a non-peptidic coupling method for the first time. The applied synthetic methodology is based on 1,3-dipolar cycloaddition chemistry of azomethine ylides and provides absolute control over the β-proline backbone stereogenic centers. An o-(trifluoromethyl)phenyl substituent contributes to appropriate stabilization of the definite acrylamide chiral cis conformation and to achieve the dipole reactivity that is not observed for aryl groups lacking strong electronegative character