The focus of this dissertation is the existence, stability, and resulting dynamical evolution of localized stationary solutions to Nonlinear Schr¨odinger (NLS) equations with periodic confining potentials in 2(+1) dimensions. I will make predictions about these properties based on a discrete lattice model of coupled ordinary differential equations with the appropriate symmetry. The latter has been justified by Wannier function expansions in a so-called tight-binding approximation in the appropriate parametric regime. Numerical results for the full 2(+1)-D continuum model will be qualitatively compared with discrete model predictions as well as with nonlinear optics experiments in optically induced photonic lattices in photorefractive crystals. The predictions are also relevant for BECs (Bose-Einstein Condensates) in optical lattices.

The reflection and transmission of an electromagnetic TE-polarized plane wave at a dielectric film between two linear semi-infinite media (substrate and cladding) is considered. All media are assumed to be homogeneous in x- and z- direction, isotropic, and non-magnetic. The permittivity of the film is assumed to be characterized by a continuously differentiable function of the transverse coordinate and the field. To obtain solutions of Maxwell´s equations that satisfy the boundary conditions the problem is reduced to a Helmholtz equation, which is transformed to a Volterra integral equation for the field intensity inside the film. The Volterra equation is solved by iteration subject to the appropriate boundary conditions. The (iteration) solutions for the linear case and for the nonlinear case are expressed in terms of a uniformly convergent series and a uniformly convergent sequence, respectively. The uniform convergence is proved using the Banach Fixed-Point Theorem. The condition for its applicability leads to a condition for the parameters of the problem. By iterating the Volterra equation an approximate solution for the intensity inside the film is presented. The mathematical basis of the procedure is outlined in detail. Using an approximate solution, the phase function,the phase shifts on reflection and transmission, the reflectivity and the absorptance are determined.Further iterations of the Volterra equation are possible.Semianalytical and numerical examples illustrate the main features of the approach. The method is succesfully applied to different permittivity functions (real, complex, Kerr-like and saturable nonlinear). The agreement between the approximate analytical solutions and numerical solutions is satisfactory. It seems that the method proposed can serve as a means to optimize certain parameters of the problem (material and/or geometrical) for particular purposes.