Mathematical Descriptions of Anisotropic Fluids and Optical Pulse Propagation

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

NSF Award Abstract - DMS-0072553 Mathematical Sciences: Mathematical Descriptions of Anisotropic Fluids and Optical Pulse Propagation Abstract 0072553 Forest The 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.