Localized and periodic exact solutions to the nonlinear Schrödinger equation with spatially modulated parameters: Linear and nonlinear lattices

Using similarity transformations we construct explicit solutions of the nonlinear Schrödinger equation with linear and nonlinear periodic potentials. We present explicit forms of spatially localized and periodic solutions, and study their properties. We put our results in the framework of the exploited perturbation techniques and discuss their implications on the properties of associated linear periodic potentials and on the possibilities of stabilization of gap solitons using polychromatic lattices.