Soliton modes, stability, and drift in optical lattices with spatially   modulated nonlinearity
  We put forward new properties of lattice solitons in materials and geometries where both, the linear refractive index and the nonlinearity are spatially modulated. We show that the interplay between linear and out-of-phase nonlinear refractive index modulations results in new soliton properties, including modifications of the soliton stability and transverse mobility, as well as shape transformations that may be controlled, e.g., by varying the light intensity. 
Yaroslav V. Kartashov, Victor A. Vysloukh, Lluis Torner