CAREER: Dispersive Waves in Nonlinear Media: Dynamics and Applications

数学科学：职业：非线性介质中的色散波：动力学与应用[摘要]库兹非线性介质中的色散波及其动力学和稳定性的研究是非线性光学和原子物理中平均场理论应用的基础。非线性的存在通常需要结合渐近和摄动方法，科学计算和严格的数学分析来实现一个坚实的数学框架，以理解给定的物理系统。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，非线性光学和原子系统的定量模型将基于第一原理开发出来。这些模型将在适当的参数制度下进行研究，简化的非线性动力系统理论可以应用。然后，结果将根据其原始实验背景进行改写，以便理论预测可以进行测试、验证和必要时修改。感兴趣的具体应用涉及光参量振荡器，光纤激光器和器件，以及玻色-爱因斯坦凝聚。所有这些系统都表现出非线性脉冲、锋面和周期波形的稳定演化。这项数学建模和分析的研究涉及三类重要问题。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实现技术的前沿。最重要的是光脉冲的稳定。本研究旨在对各种激光器和器件中非线性起关键作用的脉冲稳定性进行一般性描述。光参量振荡器在可调谐相干辐射、模式识别和光信息处理等方面具有巨大的应用潜力。这项工作将为这些光学器件中的脉冲和前端结构建立控制区域，促进该技术的实际实施。玻色-爱因斯坦凝聚体最近才在实验中实现，有望在量子逻辑和物质波输运中得到应用。捕获冷凝物并使其长时间保持是使玻色-爱因斯坦凝聚物成为可行技术的基础。本研究将聚焦于不同的周期陷阱结构，这些结构可以使凝析液稳定在吸引和排斥状态。

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.