Mathematical Descriptions of Anisotropic Fluids and Optical Pulse Propagation

NSF奖摘要-DMS-0072553数学科学：各向异性流体和光脉冲传播的数学描述摘要0072553森林各向异性流体研究项目研究在液晶聚合物等大分子流体流动中出现的关键数学问题。我们分析了描述大分子流动的张量偏微分方程组，以构造取向图案并确定其稳定性。这些分析提供了关于流动诱导相变的信息，以及在实验和制造过程中经常观察到的取向模式的模型。光脉冲传输的研究项目采用了可积系统的方法来分析脉冲在非线性光纤中的传输。控制模型方程是具有小色散的标量或耦合非线性薛定谔方程的摄动。这项研究将以前关于标量方程的结果推广到系统，研究了由耦合引起的新的不稳定性现象，构造了作为脉冲传播模型的显式解，并预测了脉冲退化的开始和命运作为光纤特性以及脉冲形状和功率的函数。拟议的研究集中在两项重要技术的核心数学问题上：高性能材料和光纤通信。许多超强材料是由大分子组成的液体生产的，大分子的形状和动力学制约着制造过程，并对材料性能负责。本研究开发了流动和微观结构相互作用的数学模型，应用这些模型来解释观察到的模式及其与材料性能的关系，并分析了影响材料加工行为和性能的其他现象。长距离光纤通信系统可以用特殊的非线性微分方程很好地描述，这些方程可以用最近发展起来的数学方法来分析。观察表明，光纤中的光脉冲通过脉冲上出现的涟漪而退化，这种现象在计算机模拟中也可以看到。该项目对纹波形成的原因进行了严格的数学理解，并开发了一种显式算法，该算法可以根据光纤和输入脉冲的特性预测脉冲退化。该工具将有助于为给定光纤设计最佳脉冲形状。

NSF Award Abstract - DMS-0072553Mathematical Sciences: Mathematical Descriptions of Anisotropic Fluids and Optical Pulse PropagationAbstract0072553 ForestThe research project on anisotropic fluids investigates critical mathematical problems arising in the flow of macromolecular fluids such as liquid crystal polymers. We analyze the tensorial partial differential equations that describe the flows of macromolecules to construct orientation patterns and determine their stability. These analyses yield information about flow-induced phase transitions and models for orientation patterns routinely observed in experiments and manufacturing processes. The research project on optical pulse propagation employs methods of integrable systems in the analysis of pulse propagation in nonlinear optical fibers. The governing model equations are perturbations of scalar or coupled nonlinear Schrodinger equations with small dispersion. The research extends previous results on scalar equations to systems, studies new instability phenomena that arise from the coupling, constructs explicit solutions that serve as models for pulse propagation, and predicts the onset and fate of pulse degradation as a function of fiber properties and of pulse shape and power. The proposed research focuses on mathematical issues central to two important technologies: high-performance materials and optical fiber communications. Many super-strong materials are produced from liquids comprised of large molecules whose shape and dynamics constrain manufacturing processes and are responsible for material properties. This research develops mathematical models for the interaction of flow and microstructure, applies these models to explain observed patterns and their relation to material properties, and analyzes other phenomena that affect processing behavior and properties of materials. Long-haul optical fiber communications systems are well-described by special nonlinear differential equations that are amenable to analysis with recently-developed mathematical methods. Observations show that light pulses in optical fibers degrade through ripples that emerge on the pulse, and this phenomenon is also seen in computer simulations. This project develops rigorous mathematical understanding of why ripples form and an explicit algorithm that predicts pulse degradation given the properties of the fiber and the input pulse. This tool will be useful to design optimal pulse shapes for given optical fibers.