Stability and Dynamics of Dispersive Waves in Nonlinear Media

非线性色散波出现在不同的应用领域。 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性质和行为至关重要。 使用跨学科的方法，结合渐近和微扰方法，科学计算和严格的数学分析与模型，这些模型是基于非线性现象的实验观察，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的波动动力学的现代方法。 特别是，各种方法减少控制方程更容易处理的偏微分方程和常微分方程系统的追求。 对于所有的原子和光学系统，我们都特别感兴趣的是局域解的稳定性和持久性，这些局域解通常是由一些基本的哈密顿（可积）系统的孤子型解产生的。局域解或脉冲的稳定性在光学和原子物理中具有根本的重要性。 在光学物理学中，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。 在过去的十年中，锁模技术已经从基础科学发展成为商业上可行的技术，并在成像，医学科学和电信中得到应用。 表征、改进和理解这些激光器的操作限制是该提案的中心焦点。此外，新兴的光子技术是基于脉冲结构的存在和稳定性，因为它们形成了全光信号处理和交换中的光比特的基础。脉冲稳定性的一个更基本的研究是在原子物理学领域，玻色-爱因斯坦凝聚已经在实验上实现了物质波的存在。 这种物质波与锁模技术有着直接的相似之处，这使得创造脉冲物质激光器成为可能。

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.