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In the present work, we revisit the issue of the self-trapping dynamical transition at a nonlinear impurity embedded in an otherwise linear lattice. For our Schrödinger chain example, we present rigorous arguments that establish necessary conditions and corresponding parametric bounds for the transition between linear decay and nonlinear persistence of a defect mode. The proofs combine a contraction mapping approach applied to the fully dynamical problem in the case of a 3D-lattice, together with variational arguments for the derivation of parametric bounds for the creation of stationary states associated with the expected fate of the self-trapping dynamical transition. The results are relevant for both power law nonlinearities and saturable ones. The analytical results are corroborated by numerical computations. The latter are performed for cases of different dimension. [ABSTRACT FROM AUTHOR]

Abstract: We examine localization of light in nonlinear (Kerr) kagome lattices in the shape of narrow strips of varying width. For the narrowest ribbon, the band structure features a flat band leading to linear dynamical trapping of an initially localized excitation. We also find a geometry-induced bistability of the nonlinear modes as the width of the strip is changed. A crossover from one to two dimensions localization behavior is observed as the width is increased, attaining two-dimensional behavior for relatively narrow ribbons. [Copyright &y& Elsevier]

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. We suggest an efficient method for creating such surface modes and the local bounded potential necessary to support the embedded modes. We demonstrate that the surface embedded modes are structurally stable, and the position of their eigenvalues inside the spectral band can be tuned continuously by adding weak nonlinearity. [ABSTRACT FROM AUTHOR]

Abstract: We study localized modes on a single Ablowitz–Ladik impurity embedded in the bulk or at the surface of a one-dimensional linear lattice. Exact expressions are obtained for the bound state profile and energy. Dynamical excitation of the localized mode reveals exponentially-high amplitude oscillations of the spatial profile at the impurity location. The presence of a surface increases the minimum nonlinearity to effect a dynamical selftrapping. [Copyright &y& Elsevier]

Abstract: We study the properties of localized impurity modes and the transmission of plane waves across a general nonlinear impurity embedded in an infinite one-dimensional linear chain. Using the formalism of the lattice Green function, we obtain nonlinear equations for the bound state energy and the transmission coefficient, for a wide class of nonlinear impurities. We specialize to a saturable nonlinear impurity, obtaining closed form expressions for the bound state energy and transmission coefficient. [Copyright &y& Elsevier]

Abstract: We discuss the formation of guided modes localized at the interface separating two different periodic photonic lattices. Employing the effective discrete model, we analyze linear and nonlinear interface modes and also predict the existence of stable interface solitons including the hybrid staggered/unstaggered lattice solitons with the tails belonging to spectral gaps of different types. [Copyright &y& Elsevier]

Abstract: We study nonlinear surface modes in two-dimensional anisotropic periodic photonic lattices and demonstrate that, in a sharp contrast to one-dimensional discrete surface solitons, the mode threshold power is lower at the surface, and two-dimensional discrete solitons can be generated easier near the lattice corners and edges. We analyze the crossover between effectively one- and two-dimensional regimes of the surface-mediated beam localization in the lattice. [Copyright &y& Elsevier]

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. We introduce a method for computing the dynamically excited frequencies, which helps us validate our stationary ansatz approach and the effective frequency concept. A general analytical estimate of the critical nonlinearity is obtained, with an extra parameter to be determined. We find this parameter to be almost constant for two-dimensional systems and prove its validity by applying it successfully to two-dimensional binary lattices. [ABSTRACT FROM AUTHOR]