Stability and Dynamics of Dispersive Waves in Nonlinear Media

非线性色散波出现在许多不同的应用领域。这些波的动力学和稳定性对于理解给定物理系统的基本物理属性和行为至关重要。使用一种跨学科的方法，将渐近和微扰方法、科学计算和严格的数学分析与基于非线性现象的实验观察的模型相结合，可以实现对特定光学和原子系统的基本理解。与建模工作相结合，数学目标是进一步发展和扩展用于量化和理解非线性色散偏微分方程波动力学的现代方法。特别是，人们寻求了各种方法来将控制方程简化为更容易处理的偏微分方程组和常微分方程组。对于这里所考虑的所有原子和光学系统来说，特别感兴趣的是定域解的稳定性和持久性，这些定域解通常是由一些潜在的哈密顿(可积)系统的孤子解产生的。定域解或脉冲的稳定性在光学和原子物理中是基本重要的。在光学物理中，脉冲解的稳定性对于确定所谓的锁模激光器的工作状态至关重要。在过去的十年里，锁模技术已经从一门基础科学发展成为一种商业上可行的技术，并在成像、医学和电信方面得到了应用。确定、改进和了解这些激光器的操作极限是该提案的中心重点。此外，新兴的光子技术是以脉冲结构的存在和稳定性为前提的，因为它们构成了全光信号处理和交换中的光比特的基础。对脉冲稳定性的更基本的研究存在于原子物理领域，在这个领域，玻色-爱因斯坦凝聚体已经通过实验实现了物质波的存在。这种物质波与锁模技术有直接的相似之处，因此有可能产生脉冲物质激光。

Nonlinear dispersive waves arise in a diverse set of application fields. The dynamics and stability of these waves is of paramount importance to understanding the underlying physical properties and behavior of a given physical system. Using an interdisciplinary approach that combines asymptotic and perturbation methods, scientific computation, and rigorous mathematical analysis with models which are based on experimental observations of nonlinear phenomena, a fundamental understanding can be achieved of specific optical and atomic systems. In conjunction with the modeling efforts, the mathematical objectives are to further develop and extend modern methods utilized for quantifying and understanding the wave dynamics of nonlinear, dispersive partial differential equations. In particular, a variety of methods for reducing the governing equations to more easily handled partial and ordinary differential equation systems is pursued. Of specific interest to all the atomic and optical systems considered here is the stability and persistence of localized solutions that often result from soliton-type solutions of some underlying Hamiltonian (integrable) system.The stability of localized solutions, or pulses, is of fundamental importance in optical and atomic physics. In optical physics, the stability of pulse solutions is critical for determining the operating regimes of the so-called, mode-locked laser. In the past decade, mode-locking technologies have gone from a fundamental science to a commercially viable technology with applications in imaging, medical sciences, and telecommunications. Characterizing, improving, and understanding the operational limits of these lasers is a central focus of the proposal. Additionally, emerging photonic technologies are predicated on the existence and stability of pulse structures as they form the basis for optical bits in all-optical signal processing and switching. A more fundamental investigation of pulse stability lies in the area of atomic physics where Bose-Einstein condensates have realized experimentally the existence of matter waves. Such matter waves have a direct analog to mode-locking technologies, allowing for the possibility of creating a pulsed matter laser.