Traveling localized waves in discrete lattices

非线性与周期性之间的相互作用是现代应用数学和非线性物理各分支研究的热点。我们的研究项目是离散和连续动力系统中局部结构的数学分析。我们的目标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的应用背景下，光子带隙工程，非线性光学，玻色-爱因斯坦凝聚体的原子物理。该研究计划包括几个具体的目标：（1）周期系统的简化模型的合理性，（2）离散格子中行波的持续性，（3）离散三维涡的分叉和稳定性。本文的研究结果将为具有规范对称性和可逆对称性的离散格点中的行波理论这一应用数学的新课题做出贡献。这一结果对许多研究组在理论和实验上研究光囚禁玻色-爱因斯坦凝聚体、光折变晶格和耦合光波导等方面具有重要意义。

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.