Localised modes in discrete lattices

Interplay between nonlinearity and periodicity is the focus of recent studies in different branches of modern applied mathematics and nonlinear physics. Our research program is in mathematical analysis of discrete dynamical systems. We aim to study nonlinear models expressed by the differential advanced-delay equations, difference equations and partial differential equations in the context of applications to photonic band-gap engineering, nonlinear optics, and atomic physics of Bose-Einstein condensates. The research program consists of several specific goals (problems): (1) derivation of the discrete nonlinear Schrodinger equation and the coupled-mode (Dirac) system, (2) persistence of traveling waves in discrete lattices, and (3) bifurcations and stability of three-dimensional discrete vortices.