Nonlinear surface impurity in a semi-infinite lattice

Abstract

We examine the formation of bound states on a generalized nonlinear impurity located at or near the beginning (surface) of a linear, tight-binding semi-infinite lattice. Using the formalism of lattice Green functions, we obtain in closed form the number of bound states as well as their energies and probability profiles, for different nonlinearity parameter values and nonlinearity exponents, at different distances from the surface. It is shown that close to the surface, the amount of nonlinearity needed to create a bound state or to effect dynamical selftrapping, increases (decreases) depending on whether the exponent is smaller (larger) than, approximately, two.

Discrete surface solitons in two dimensions

“…Recently, a new implementation of surface solitary waves was proposed [4] and experimentally created [5] in nonlinear optics, in the form of discrete localized pulses supported at the edge of a semi-infinite array of nonlinear waveguides. Twocomponent surface solitons were analyzed too [6], and it was predicted that solitons may be supported at an edge of a discrete chain by a nonlinear impurity [7]. Parallel to that, surface gap solitons were predicted [8] and created in an experiment [9] at an edge of a waveguiding array built into in a self-defocusing continuous medium.…”

Decay of a nonlinear impurity in a structured continuum from a nonlinear Fano-Anderson model

“…Such a result is well known and has been widely studied, for instance, in tight-binding lattice models with nonlinear impurity modes. 26,34,35 Stability of these nonlinear bound states has also been discussed for some specific models ͑see, for instance, Ref. 26͒.…”

“…20-31 and references therein͒. A few relevant examples of nonlinear effects that may accompany tunneling processes are atom-atom interaction in dense matter wave systems, [20][21][22] polaronic behavior in strongly coupled electron-vibration systems, [23][24][25][26] or material nonlinearity in coupled optical waveguides or photonic crystal resonators. [27][28][29][30][31] While the decay dynamics of an unstable state in a linear system, including the validity of the golden rule in the 2 t limit and the occurrence of nonexponential deviations at short and long times, is to date well understood, the effects of nonlinearity on the decay dynamics have not received an adequate attention yet.…”

“…Specifically, we consider the decay dynamics of a nonlinear discrete state embedded into a structured linear continuum in the framework of a nonlinear extension of the Fano-Anderson model, 32,33 which may describe electron or photon transport in one-dimensional lattices with a local nonlinear impurity mode. 25,26,29,30,34,35 By means of a multiple-scale asymptotic analysis, a decay law for the survival probability P͑t͒ is derived which provides an extension to the nonlinear case of the 2 t Van Hove limit. 2,5,6 In the linear regime, the 2 t limit yields a purely exponential decay ͑golden rule͒; the nonlinearity introduces nonexponential features which should be observable whenever the nonlinear-induced shift of the impurity mode resonance spans a spectral region where the reservoir spectral function changes rapidly.…”

Observation of Defect-Free Surface Modes in Optical Waveguide Arrays

“…In contrast, a surface defect is required for surface waves to exist in the photonic lattices composed of identical straight waveguides. It was found that Tamm surface waves can exist at the edge of an array of optical waveguides when the effective refractive index of the boundary waveguide is modified above a certain threshold [17][18][19][20][21][22][23].…”