Solitons in combined linear and nonlinear lattice potentials

We study ordinary solitons and gap solitons (GS's) in the framework of the one-dimensional Gross-Pitaevskii equation (GPE) with a combination of both linear and nonlinear lattice potentials. The main points of the analysis are the effects of (in)commensurability between the lattices, the development of analytical methods, viz., the variational approximation (VA) for narrow ordinary solitons and various forms of the averaging method for broad solitons of both types, and also the study of the mobility of the solitons. Under the direct commensurability (equal periods of the lattices, Llin=Lnonlin), the family of ordinary solitons is similar to its counterpart in the GPE without external potentials. In the case of the subharmonic commensurability with Llin=(1/2)Lnonlin, or incommensurability, there is an existence threshold for the ordinary solitons and the scaling relation between their amplitude and width is different from that in the absence of the potentials. GS families demonstrate a bistability unless the direct commensurability takes place. Specific scaling relations are found for them as well. Ordinary solitons can be readily set in motion by kicking. GS's are also mobile and feature inelastic collisions. The analytical approximations are shown to be quite accurate, predicting correct scaling relations for the soliton families in different cases. The stability of the ordinary solitons is fully determined by the Vakhitov-Kolokolov (VK) criterion (i.e., a negative slope in the dependence between the solitons's chemical potential μ and norm N). The stability of GS families obeys an inverted ("anti-VK") criterion dμ/dN>0, which is explained by the approximation based on the averaging method. The present system provides for the unique possibility to check the anti-VK criterion, as μ(N) dependencies for GS's feature turning points except in the case of direct commensurability.