Mathematical Descriptions of Anisotropic Fluids and Optical Pulse Propagation

数学科学：各向异性流体和光脉冲传播的数学描述[Abstract] [0072553 forest]各向异性流体研究项目研究液晶聚合物等大分子流体流动中出现的关键数学问题。我们分析描述大分子流动的张量偏微分方程来构建取向模式并确定其稳定性。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模型。光脉冲传播研究项目采用可积系统的方法分析脉冲在非线性光纤中的传播。控制模型方程是小色散的标量扰动或耦合非线性薛定谔方程。该研究将以往关于标量方程的结果扩展到系统，研究了由耦合引起的新的不稳定现象，构建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结局，作为光纤特性和脉冲形状和功率的函数。提出的研究重点是两个重要技术的核心数学问题：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，这些大分子的形状和动力学限制了制造过程，并决定了材料的性能。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，并分析影响材料加工行为和性能的其他现象。长途光纤通信系统可以用特殊的非线性微分方程很好地描述，这些方程可以用最新发展的数学方法进行分析。观察表明，光纤中的光脉冲通过脉冲上出现的波纹而衰减，这一现象在计算机模拟中也可以看到。该项目对波纹形成的原因进行了严格的数学理解，并给出了一个明确的算法，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的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.