Standard models of periodically modulated nonlinear media, such as photonic crystals and Bose-Einstein condensates (BECs) trapped in optical lattices (OLs), are often described by the nonlinear Schroedinger/Gross-Pitaevskii equations with periodic potentials. We consider a model including such a periodic potential and the attractive or repulsive nonlinearity concentrated at a single point or at a set of two points, which are represented by delta-functions. For the attractive or repulsive nonlinearity, the model gives rise to ordinary solitons or gap solitons (GSs). These localized modes reside, respectively, in the semi-infinite gap, or finite bandgaps of the system's linear spectrum. The solitons are pinned to the delta-functions. Realizations of these models are relevant to optics and BECs. We demonstrate that the single nonlinear deltafunction supports families of stable ordinary solitons and GSs in the cases of the selfattractive and repulsive nonlinearity, respectively. We also show that the delta-function can support stable GSs in the first finite bandgap in the case of the self-attraction. The stability of the GSs in the second finite bandgap is investigated too. In addition to the numerical analysis, analytical approximations are developed for the solitons in the semi-infinite gap and two lowest finite bandgaps. In the model with the symmetric pair of delta-functions, we investigate the effect of the spontaneous symmetry breaking of the pinned solitons.
|Title of host publication||Optical Lattices|
|Subtitle of host publication||Structures, Atoms and Solitons|
|Publisher||Nova Science Publishers, Inc.|
|Number of pages||36|
|State||Published - 2011|