PHYSICAL REVIEW A 87, 063849 (2013) Bright, dark, antidark, and kink solitons in media with periodically alternating sign of nonlinearity Yannis Kominis School of Electrical and Computer Engineering, National Technical University of Athens, Zographou GR-15773, Greece and Department of Mathematics, University of Patras, Patras GR-26500, Greece (Received 13 May 2013; published 28 June 2013) The formation of solitary waves (SW) in media with periodically alternating sign of nonlinearity is investigated under a geometrical phase space approach. It is shown that a remarkably rich set of all types of SW, including bright, dark, antidark, and kink solitons is supported by this type of structure. The existence conditions of all SW are systematically defined in the parameter space of the system and their propagation dynamics are investigated through numerical simulations. DOI: 10.1103/PhysRevA.87.063849 PACS number(s): 42.65.Tg, 42.65.Jx, 42.65.Sf, 03.75.Lm I. INTRODUCTION Spatial self-localization of waves in nonlinear periodic structures is a ubiquitous phenomenon resulting from the interplay between the medium inhomogeneity and nonlinearity that has no analog in homogeneous media. Its effect gives rise to the existence of a rich set of solitary waves (SW) in a large variety of physical systems. Among them, the formation and propagation of light SW in photonic structures such as optical lattices and waveguide arrays are widely studied in the context of nonlinear optics [1–4], whereas matter SW are studied in the context of Bose-Einstein condensates (BEC) [5–8], with theoretical studies on both fields progressing in parallel due to common underlying models of wave propagation. In general, SW have the form of a spatially localized transi- tion between two asymptotic states. Different types of SW can be characterized on the basis of their asymptotic states, with bright solitons having zero asymptotic states, dark (antidark) solitons having nonzero asymptotic states of opposite (same) sign, and kink solitons having asymptotic states with different values in general. The existence of such SW is closely related to complex dynamics of the system describing the spatial wave profile formation with the respective asymptotic states having the form of hyperbolic plane-wave solutions of the system. From a geometrical point of view, in the phase space of the system, SW can be found as intersections of the invariant stable and unstable manifolds of the hyperbolic solutions corresponding to their asymptotic states [9]. Among the different types of SW, bright solitons have been most extensively theoretically studied and experimentally demonstrated in several configurations with varying degree of spatial complexity ranging from monochromatic modulation of the linear refractive index to superlattices and nonperiodic structures. Due to the presence of the inhomogeneity, bright solitons can be formed in either focusing or defocusing media, in contrast to the homogeneous case [10]. Theoretical studies have been based either on continuous models consisting of the nonlinear Schr¨ odinger (NLS) equation with spatially varying coefficients [11–13] or simplified discrete models under the tight-binding approximation describing deeply trapped narrow SW [4]. Dark solitons have been theoretically studied in both discrete [14] and continuous models [15] and their existence has been shown in various configurations. Antidark as well as kink solitons have been less intensively studied than dark and bright solitons. Antidark solitons have been theoretically studied in continuous vector two-component models describing either weakly trapped SW in band gap photonic structures with the utilization of coupled mode theory [16,17] or in binary Bose-Einstein condensates [18,19]. Discrete binary models describing waveguide arrays with alternating positive-negative couplings have also been shown to support antidark soliton solutions [20,21]. Kink solitons (also referred to as shocks or domain walls) in periodic structures have been considered mostly in discrete models [21–23] as well as in coupled wave equations describing vector SW [24,25]. The formation of surface SW having the form of kink solitons has also been studied in nonperiodic structures consisting either of interfaces between lattices [26–28] or localized modulations of the medium [29,30]. In the latter case the SW profile asymmetry is directly enforced by the nonperiodic spatial modulation of the medium in contrast to the former case of periodic modulations where the profile asymmetry results from the dynamical interplay between the nonlinearity and the periodicity of the medium. Structures of increasing complexity of the spatial mod- ulation of the medium have also been considered from the point of view of engineering the inhomogeneity in order to provide desirable properties of SW formation and propagation dynamics. Among them, a large amount of work has been focused on media where, in addition to the linear properties, the nonlinearity is also transversely modulated forming nonlinear lattices [13,31,32]. In such structures both the strength and the sign of the nonlinearity can vary across their transverse dimension. A periodic alternation of the sign of the nonlinear refractive index corresponds to a periodically focusing and defocusing optical medium in the context of nonlinear optics, whereas in the context of BEC the periodic alternation of the sign of the scattering length corresponds to periodically repulsive and attractive interactions between the atoms [32]. Layered structures combining piecewise constant profiles of the linear and nonlinear properties of the medium are usually referred to as Kronig-Penney (KP) lattices and have been considered in terms of SW formation and propagation [33–35]. In this work, a layered medium with periodically alternating sign of nonlinearity is considered. The continuous model of the NLS equation with piecewise constant coefficients of the linear and nonlinear terms is utilized to describe wave propagation under no restriction as those implied by the tight-binding approximation or the coupled-mode approach. A geometrical phase space analysis reveals the remarkable richness of all 063849-1 1050-2947/2013/87(6)/063849(9) ©2013 American Physical Society