Mathematical Descriptions of Anisotropic Fluids and Optical Pulse Propagation

美国国家科学基金会奖摘要- DMS-0072553数学科学：各向异性流体和光脉冲调制的数学描述摘要0072553森林各向异性流体的研究项目调查在大分子流体如液晶聚合物的流动中出现的关键数学问题。 我们分析了张量偏微分方程，描述了大分子的流动，构建取向模式，并确定其稳定性。 这些分析产生的信息流诱导的相变和模型的取向模式，在实验和制造过程中经常观察到的。 光脉冲在非线性光纤中传输的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微扰。 该研究扩展了以前的结果标量方程系统，研究新的不稳定现象，产生的耦合，构建明确的解决方案，作为脉冲传播的模型，并预测脉冲退化的发生和命运的函数的纤维性能和脉冲形状和功率。拟议的研究重点是两项重要技术的核心数学问题：高性能材料和光纤通信。 许多超强材料都是由大分子组成的液体制成的，这些大分子的形状和动力学限制了制造过程，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是由特殊的非线性微分方程描述的，这些方程可以用最近发展起来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退化，这种现象也在计算机模拟中看到。 该项目对波纹形成的原因进行了严格的数学理解，并提出了一种明确的算法，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.