Solitons in nonlinear lattices

Abstract

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by purely nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are surveyed too, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation (BEC). The solitons are considered in one, two, and three dimensions (1D, 2D, and 3D). Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of 1D solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for the theoretical and experimental studies alike. In both the 1D and 2D cases, the mechanism which creates solitons in NLs is principally different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.Peer ReviewedPostprint (published version

One- and two-dimensional solitons supported by singular modulation of quadratic nonlinearity

“…In that case, the strongly concentrated nonlinearity may be described by the delta-function [9,33,35]. Another variety of the strongly localized self-focusing χ (3) nonlinearity is a discrete linear lattice with one [10] or two nonlinear [11] sites embedded into it, or externally coupled to the lattice [12].…”

“…Most works on this topic have been dealing with the cubic, alias χ (3) , nonlinearity. In optical media, spatially nonuniform Kerr nonlinearity can be induced by an accordingly designed inhomogeneous density of nonlinearity-inducing dopants [4], by an inhomogeneous distribution of detuning in a uniform resonant-dopant density [3], or in composite media assembled of different materials [5]. Similar nonlinearity landscapes are relevant in models of Bose-Einstein condensates, where virtually any landscape, controlled by the optical Feshbach resonance [6], can be "painted" in space by a rapidly moving laser beam [7].…”

“…Recently, much attention has also been attracted to the creation of effective nonlinear potentials (alias "pseudopotentials", as they are called in solid-state physics [2]) by means of spatial modulation of local strength of the nonlinearity [3]. Most works on this topic have been dealing with the cubic, alias χ (3) , nonlinearity.…”

“…Recently, much attention has also been attracted to the creation of effective nonlinear potentials (alias "pseudopotentials", as they are called in solid-state physics [2]) by means of spatial modulation of local strength of the nonlinearity [3]. Most works on this topic have been dealing with the cubic, alias χ (3) , nonlinearity. In optical media, spatially nonuniform Kerr nonlinearity can be induced by an accordingly designed inhomogeneous density of nonlinearity-inducing dopants [4], by an inhomogeneous distribution of detuning in a uniform resonant-dopant density [3], or in composite media assembled of different materials [5].…”

“…In the same context, the nonlinearity can be patterned by means of the magnetic Feshbach resonance, using magnetic lattices [8]. The limit case of the spatially inhomogeneous χ (3) self-interaction in the one-dimensional (1D) geometry corresponds to a very narrow strip with strong nonlinearity, which is embedded into a linear host medium. In that case, the strongly concentrated nonlinearity may be described by the delta-function [9,33,35].…”