Traveling localized waves in discrete lattices

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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 localized structures in discrete and continuous 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: (1) justification of the reduced models for periodic systems, (2) persistence of traveling waves in discrete lattices, (3) bifurcations and stability of discrete three-dimensional vortices. The results of the proposed research will contribute towards a new topic in applied mathematics that is the theory of traveling waves in discrete lattices with the gauge and reversibility symmetries. The outcomes will be significant to many research groups that study theoretically and experimentally optically trapped Bose-Einstein condensates, photorefractive crystal lattices, and coupled optical waveguides.