Traveling localized waves in discrete lattices

非线性和周期性之间的相互作用是现代应用数学和非线性物理不同分支最近研究的焦点。我们的研究项目是离散和连续动力系统中局域结构的数学分析。我们的目标是在玻色-爱因斯坦凝聚体的光子带隙工程、非线性光学和原子物理的应用背景下，研究由微分超前延迟方程、差分方程和偏微分方程组表示的非线性模型。研究程序包括几个具体的目标：(1)证明周期系统的简化模型，(2)离散格子中行波的持久性，(3)离散三维涡旋的分叉和稳定性。所提出的研究结果将有助于应用数学中的一个新课题，即具有规范对称性和可逆性对称性的离散晶格中的行波理论。这一结果将对许多从理论和实验上研究玻色-爱因斯坦凝聚体、光折变晶格和耦合光波导的研究小组具有重要意义。

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.