Vortex ring beams in nonlinear -symmetric systems.

In this paper, we investigate a two-dimensional photonic array featuring a circular shape and an alternating gain and loss pattern. Our analysis revolves around determining the presence and resilience of optical ring modes with varying vorticity values. This investigation is conducted with respect to both the array's length and the strength of the non-Hermitian parameter.

Fractional nonlinear electrical lattice.

We examine the linear and nonlinear modes of a one-dimensional nonlinear electrical lattice, where the usual discrete Laplacian is replaced by a fractional discrete Laplacian. This induces a long-range intersite coupling that, at long distances, decreases as a power law. In the linear regime, we compute both the spectrum of plane waves and the mean-square displacement (MSD) of an initially localized excitation, in closed form in terms of regularized hypergeometric functions and the fractional exponent.

Fractional discrete vortex solitons.

We examine the existence and stability of nonlinear discrete vortex solitons in a square lattice when the standard discrete Laplacian is replaced by a fractional version. This creates a new, to the best of our knowledge, effective site-energy term, and a coupling among sites, whose range depends on the value of the fractional exponent $\alpha$, becoming effectively long range at small $\alpha$ values. At long distance, it can be shown that this coupling decreases faster than exponentially: $\sim\exp (- |{\textbf{n}}|)/\sqrt {|n|}$.

The fractional nonlinear [Formula: see text] dimer.

We examine a fractional discrete nonlinear Schrodinger dimer, where the usual first-order derivative in the time evolution is replaced by a non integer-order derivative. The dimer is nonlinear (Kerr) and [Formula: see text]-symmetric, and for localized initial conditions we examine the exchange dynamics between both sites. By means of the Laplace transformation technique, the linear [Formula: see text] dimer is solved in closed form in terms of Mittag-Leffler functions, while for the nonlinear regime, we resort to numerical computations using the direct explicit Grunwald algorithm.

Localized vortex beams in anisotropic Lieb lattices.

We address the issue of nonlinear modes in a two-dimensional waveguide array, spatially distributed in the Lieb lattice geometry, and modeled by a saturable nonlinear Schrödinger equation. In particular, we analyze the existence and stability of vortex-type solutions finding localized patterns with symmetric and asymmetric profiles, ranging from topological charge =1 to =3. By taking into account the presence of anisotropy, which is inherent to experimental realization of waveguide arrays, we identify different stability behaviors according to their topological charge.

Seltrapping in flat band lattices with nonlinear disorder.

We study the transport properties of an initially localized excitation in several flat band lattices, in the presence of nonlinear (Kerr) disorder. In the weak nonlinearity regime, the dynamics is controlled by the degeneracy of the bands leading to a linear form of selftrapping. In the strong nonlinearity regime, the dynamics of the excitations depends strongly on the local environment around the initial excitation site that leads to a highly fluctuating selfrapping profile.

Nonlinear optical lattices with a void impurity.

We examine a one-dimensional nonlinear (Kerr) waveguide array which contains a single "void" waveguide where the nonlinearity is identically zero. We uncover a family of nonlinear localized modes centered at or near the void, and their stability properties. Unlike a usual impurity problem, here the void acts like a repulsive impurity causing the center of the simplest mode to lie to the side of the void's position.

Topological properties of a bipartite lattice of domain wall states.

We propose a generalization of the Su-Schrieffer-Heeger (SSH) model of the bipartite lattice, consisting of a periodic array of domain walls. The low-energy description is governed by the superposition of localized states at each domain wall, forming an effective mono-atomic chain at a larger scale. When the domain walls are dimerized, topologically protected edge states can appear, just like in the original SSH model.

Disorder-free weak dynamic localization in deformable lattices.

We study the electron transport in a deformable lattice modeled in the semiclassical approximation as a discrete nonlinear elastic chain where acoustic phonons are in thermal equilibrium at temperature T. We reveal that an effective dynamic disorder induced in the system due to thermalized phonons is not strong enough to produce Anderson localization. However, for weak nonlinearity we observe a transition between ballistic (low T) and diffusive (high T) regimes, while for strong nonlinearity the transition occurs between the localized soliton (low T) and diffusive (high T) regimes.

Solitons in a modified discrete nonlinear Schrödinger equation.

We study the bulk and surface nonlinear modes of a modified one-dimensional discrete nonlinear Schrödinger (mDNLS) equation. A linear and a modulational stability analysis of the lowest-order modes is carried out. While for the fundamental bulk mode there is no power threshold, the fundamental surface mode needs a minimum power level to exist.

Compact modes in quasi one dimensional coupled magnetic oscillators.

In this work we study analytically and numerically the spectrum and localization properties of three quasi-one-dimensional (ribbons) split-ring resonator arrays which possess magnetic flatbands, namely, the stub, Lieb and kagome lattices, and how their spectra are affected by the presence of perturbations that break the delicate geometrical interference needed for a magnetic flatband to exist. We find that the stub and Lieb ribbons are stable against the three types of perturbations considered here, while the kagome ribbon is, in general, unstable. When losses are incorporated, all flatbands remain dispersionless but become complex, with the kagome ribbon exhibiting the highest loss rate.

Transport of localized and extended excitations in chains embedded with randomly distributed linear and nonlinear n-mers.

We examine the transport of extended and localized excitations in one-dimensional linear chains populated by linear and nonlinear symmetric identical n-mers (with n=3, 4, 5, and 6), randomly distributed. First, we examine the transmission of plane waves across a single linear n-mer, paying attention to its resonances, and looking for parameters that allow resonances to merge. Within this parameter regime we examine the transmission of plane waves through a disordered and nonlinear segment composed by n-mers randomly placed inside a linear chain.

Observation of Localized States in Lieb Photonic Lattices.

We present the first experimental demonstration of a new type of localized state in the continuum, namely, compacton-like linear states in flat-band lattices. To this end, we employ photonic Lieb lattices, which exhibit three tight-binding bands, with one being perfectly flat. Discrete predictions are confirmed by realistic continuous numerical simulations as well as by direct experiments.

Bounded dynamics in finite PT-symmetric magnetic metamaterials.

We examine the PT-symmetry-breaking transition for a magnetic metamaterial of a finite extent, modeled as an array of coupled split-ring resonators in the equivalent circuit model approximation. Small-size arrays are solved completely in closed form, while for arrays larger than N=5 results were computed numerically for several gain and loss spatial distributions. In all cases, it is found that the parameter stability window decreases rapidly with the size of the array, until at N=20 approximately it is not possible to support a stable PT-symmetric phase.

Self-trapping transition in nonlinear cubic lattices.

We explore the fundamental question of the critical nonlinearity value needed to dynamically localize energy in discrete nonlinear cubic (Kerr) lattices. We focus on the effective frequency and participation ratio of the profile to determine the transition into localization in one-, two-, and three-dimensional lattices. A simple and general criterion is developed, for the case of an initially localized excitation, to define the transition region in parameter space ("dynamical tongue") from a delocalized to a localized profile.

Enhanced distribution of a wave-packet in lattices with disorder and nonlinearity.

We show, numerically and experimentally, that the presence of weak disorder results in an enhanced energy distribution of an initially localized wave-packet, in one- and two-dimensional finite lattices. The addition of a focusing nonlinearity facilitates the spreading effect even further by increasing the wave-packet effective size. We find a clear transition between the regions of enhanced spreading (weak disorder) and localization (strong disorder).

Surface bound states in the continuum.

We introduce a novel concept of surface bound states in the continuum, i.e., surface modes embedded into the linear spectral band of a discrete lattice.

Discrete and surface solitons in photonic graphene nanoribbons.

We analyze localization of light in honeycomb photonic lattices restricted in one dimension, which can be regarded as an optical analog of graphene nanoribbons. We discuss the effect of lattice topology on the properties of discrete solitons excited inside the lattice and at its edges. We discuss a type of soliton bistability, geometry-induced bistability, in the lattices of a finite extent.

Observation of localized modes at phase slips in two-dimensional photonic lattices.

We experimentally study light localization at phase-slip waveguides and at the intersection of phase slips in a two-dimensional (2D) square photonic lattice. Such systems allow for the observation of a variety of effects, including the existence of spatially localized modes for low powers, the generation of strongly localized states in the form of discrete bulk and surface solitons, as well as a crossover between one-dimensional and 2D localization.

Fano resonances in waveguide arrays with saturable nonlinearity.

We study a waveguide array with an embedded nonlinear saturable impurity. We solve the impurity problem in closed form and find the nonlinear localized modes. Next, we consider the scattering of a small-amplitude plane wave by a nonlinear impurity mode, and discover regions in parameter space where transmission is fully suppressed.

Interface solitons in two-dimensional photonic lattices.

We analyze localization of light at the interfaces separating square and hexagonal photonic lattices, as recently realized experimentally for two-dimensional laser-written waveguides in silica glass with self-focusing nonlinearity [Opt. Lett.33, 663 (2008)].

Nonlinear localized modes at phase-slip defects in waveguide arrays.

We study light localization at a phase-slip defect created by two semi-infinite mismatched identical arrays of coupled optical waveguides. We demonstrate that the nonlinear defect modes possess the specific properties of both nonlinear surface modes and discrete solitons. We analyze the stability of the localized modes and their generation in both linear and nonlinear regimes.

Surface solitons in chirped photonic lattices.

We study surface modes at the edge of a semi-infinite chirped photonic lattice in the framework of an effective discrete nonlinear model. We demonstrate that the lattice chirp can change dramatically the conditions for the mode localization near the surface, and we find numerically the families of discrete surface solitons in this case. Such solitons do not require any minimum power to exist provided the chirp parameter exceeds some critical value.

Localized modes and bistable scattering in nonlinear network junctions.

We study the properties of junctions created by the crossing of N identical branches of linear discrete networks. We reveal that for N>2 such a junction creates a topological defect and supports two types of spatially localized modes. We analyze the wave scattering by the junction defect and demonstrate nonzero reflection for any set of parameters.

Observation of surface gap solitons in semi-infinite waveguide arrays.

We report on the observation of surface gap solitons found to exist at the interface between uniform and periodic dielectric media with defocusing nonlinearity. We demonstrate strong self-trapping at the edge of a LiNbO3 waveguide array and the formation of staggered surface solitons with propagation constant inside the first photonic band gap. We study the crossover between linear repulsion and nonlinear attraction at the surface, revealing the mechanism of nonlinearity-mediated stabilization of the surface gap modes.