CAREER: Dispersive Waves in Nonlinear Media: Dynamics and Applications

NSF奖摘要- DMS-0092682数学科学：职业：非线性介质中的色散波：动力学和应用摘要0092682 Kutz非线性介质中色散波及其动力学和稳定性的研究是原子物理学中非线性光学和平均场理论应用的基础。 非线性的存在通常需要结合渐进和扰动方法、科学计算和严格的数学分析，以获得理解给定物理系统的坚实数学框架。 通过与工业合作伙伴以及物理和电气工程学术界成员的直接合作，将根据第一原理开发感兴趣的非线性光学和原子系统的定量模型。 这些模型将研究在适当的参数制度，简化的非线性动力系统理论可以应用。然后将根据原始实验背景对结果进行重新设计，以便理论预测可以进行测试，验证和必要的修改。 感兴趣的具体应用涉及光学参量振荡器，光纤激光器和设备，玻色-爱因斯坦凝聚。 所有这些系统都表现出非线性脉冲、波前和周期性波列的稳定演化。 数学建模与分析的研究主要涉及三类重要问题。 随着材料和器件的快速发展，非线性光学仍然处于通信和信息系统技术的最前沿。最重要的是光脉冲的稳定。 这项研究的目的是提供一个一般性的描述，在各种激光器和设备的非线性起着关键作用的脉冲的稳定性。 光学参量振荡器在可调谐相干辐射、模式识别和光学信息处理等方面具有巨大的应用潜力。 这项工作将建立这些光学器件中脉冲和前沿结构的控制区域，促进该技术的实际实施。 玻色-爱因斯坦凝聚，最近才在实验上实现，预计将在量子逻辑和物质波传输中应用。 捕获凝聚体并使其长时间维持是使玻色-爱因斯坦凝聚体成为可行技术的基础。这项研究将集中在各种周期性的陷阱配置，可以稳定的凝聚在吸引和排斥状态。

NSF Award Abstract - DMS-0092682Mathematical Sciences: CAREER: Dispersive Waves in Nonlinear Media: Dynamics and ApplicationsAbstract0092682 KutzThe study of dispersive waves and their dynamics and stability in nonlinear media is fundamental in applications arising from nonlinear optics and mean-field theories in atomic physics. The presence of nonlinearity often requires a combination of asymptotic and perturbation methods, scientific computation, and rigorous mathematical analysis to achieve a solid mathematical framework for understanding a given physical system. By direct collaboration with both industrial partners and members of the physics and electrical engineering academic community, quantitative models for the nonlinear optics and atomic systems of interest will be developed based upon first principles. These models will be studied in appropriate parameter regimes where simplified nonlinear dynamical systems theory can be applied. The results will then be recast in terms of their original experimental context so that the theoretical predictions can be tested, verified, and modified as necessary. The specific applications of interest concern optical parametric oscillators, optical fiber lasers and devices, and Bose-Einstein condensates. All these systems exhibit a stable evolution of nonlinear pulses, fronts, and periodic wavetrains. This research in mathematical modeling and analysis addresses three classes of important problems. With rapidly developing materials and devices, nonlinear optics remains at the forefront of enabling technologies for communications and information systems. Of primary importance is the stabilization of optical pulses. This research aims to provide a general description of the stability of pulses in a wide variety of lasers and devices where nonlinearity plays a key role. Optical parametric oscillators have tremendous potential application for tunable coherent radiation, pattern recognition, and optical information processing. This work will establish regions of control for the pulse and front structures in these optical devices, facilitating practical implementation of the technology. Bose-Einstein condensates, which have been only recently realized experimentally, are expected to have applications in quantum logic and matter-wave transport. Trapping the condensate and sustaining it for long time periods are fundamental for making the Bose-Einstein condensates a viable technology. This research will focus on various periodic trap configurations that can stabilize the condensate in both attractive and repulsive states.