Nonlinear optics and light localization in periodic photonic lattices

We review the recent developments in the field of photonic lattices emphasizing their unique properties for controlling linear and nonlinear propagation of light. We draw some important links between optical lattices and photonic crystals pointing towards practical applications in optical communications and computing, beam shaping, and bio-sensing.Comment: to appear in Journal of Nonlinear Optical Physics & Materials (JNOPM

Solitary waves in the Nonlinear Dirac Equation

In the present work, we consider the existence, stability, and dynamics of solitary waves in the nonlinear Dirac equation. We start by introducing the Soler model of self-interacting spinors, and discuss its localized waveforms in one, two, and three spatial dimensions and the equations they satisfy. We present the associated explicit solutions in one dimension and numerically obtain their analogues in higher dimensions. The stability is subsequently discussed from a theoretical perspective and then complemented with numerical computations. Finally, the dynamics of the solutions is explored and compared to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger equation. A few special topics are also explored, including the discrete variant of the nonlinear Dirac equation and its solitary wave properties, as well as the PT-symmetric variant of the model

On Origin and Dynamics of the Discrete NLS Equation

"We investigate soliton-like dynamics in the descrete nonlinear Schroedinger equation (DNLSE) describing the generic 3-element descrete nonlinear system with a dispersion. The DNLSE (1+2) is solved on the 3 x N descrete lattice, where N is the variable number introduced through the descretized dispersion term. In quasi-linear and strongly nonlinear regimes the evolution shows robustness with respect to the N variation. However, the intermediate regime often exhibiting chaos, appears highly sensitive to the number of descrete points, making an exact solving of the DNLSE (1+2) a dubious task. We briefly outline implications on other continuum models alike the NLSE.

Modulation Instability in Two-dimensional Nonlinear Schrodinger Lattice Models with Dispersion and Long-range Interactions

"The problem of modulation instability of continuous wave and array soliton solutions in the framework of a two-dimensional continuum-discrete nonlinear Schrodinger lattice model which accounts for dispersion and ling-range interactIONS BETWEEN ELEMENTS, IS INVESTIGATED. APPLICATION OF THE LINEAR STABILITY ANALYSIS BASED ON AN ENERGETIC PRINCIPLE AND A VARIATIONAL APPROACH, WHICH WERE ORIGINALLY DEVELOPED FOR THE CONTINUUM NONLINEAR SCHRODINGER MODEL, IS PROPOSED. Analytical expressions for the corresponding instability thresholds and the growth rate spectra are calculated.

Solitons in the discrete nonpolynomial Schrodinger equation

We introduce a species of the discrete nonlinear Schrodinger (DNLS) equation, which is a model for a self-attractive Bose-Einstein condensate confined in a combination of a cigar-shaped trap and deep optical lattice acting in the axial direction. The equation is derived as a discretization of the respective nonlinear nonpolynomial Schrodinger equation. Unlike previously considered varieties of one-dimensional DNLS equations, the present discrete model admits on-site collapse. We find two families of unstaggered on-site-centered discrete solitons, stable and unstable ones, which include, respectively, broad and narrow solitons, their stability exactly complying with the Vakhitov-Kolokolov criterion. Unstable on-site solitons either decay or transform themselves into robust breathers. Intersite-centered unstaggered solitons are unstable to collapse; however, they may be stabilized by the application of a sufficiently strong kick, which turns them into moving localized modes. Persistently moving solitons can be readily created too by the application of the kick to stable on-site unstaggered solitons. In the same model, staggered solitons, which are counterparts of gap solitons in the continuum medium, are possible if the intrinsic nonlinearity is self-repulsive. All on-site staggered solitons are stable, while intersite ones have a small instability region. The staggered solitons are immobile