Traveling localized waves in discrete lattices

非线性与周期性之间的相互作用是近年来现代应用数学和非线性物理各个分支研究的热点。我们的研究方向是离散和连续动力系统中局部结构的数学分析。我们的目的是研究由微分超前延迟方程、差分方程和偏微分方程表示的非线性模型在光子带隙工程、非线性光学和玻色-爱因斯坦凝聚体原子物理中的应用。该研究计划包括几个具体目标：(1)周期系统简化模型的证明；(2)离散格中行波的持久性；(3)离散三维涡旋的分岔和稳定性。本文的研究结果将有助于建立具有规范对称性和可逆性对称性的离散格行波理论这一应用数学的新课题。这一结果将对许多研究小组在理论和实验上研究光捕获玻色-爱因斯坦凝聚物、光折变晶格和耦合光波导具有重要意义。

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.