Two-dimensional solitons and vortices in media with incommensurate linear and nonlinear lattice potentials

We construct families of ordinary and gap solitons (GSs), including solitary vortices, in the two-dimensional (2D) system based on the nonlinear-Schr\"Aodinger/Gross-Pitaevskii equation with the 2D or quasi-1D (Q1D) periodic linear potential, combined with the periodic modulation of the cubic nonlinearity (also in the 2D or Q1D form), which is, generally, incommensurate with the linear potential, thus forming a \nonlinear quasicrystal". Stable vortices are built as complexes of four peaks with the separation between them equal to the double period of the linear potential. The system may be realized in photonic crystals or Bose-Einstein condensates (BECs). The variational approximation (VA) is applied to ordinary solitons (residing in the semi-infinite gap), and numerical methods are used to construct solitons of all the types. Stability regions are identified for soliton families in all the versions of the model.