Mathematical Sciences: Geometric Models and Methods in Nonlinear Optics

在这里提出的项目中，PI打算集中在几何方法发挥基本作用的两个特定领域。这里应用的方法是：第一，孤子和近可积系统理论；第二，几何奇点理论。第一个是关于色散差分格式的建模。非线性光学方程经常在色散效应比黏性效应更占优势的情况下运行。一个特别相关的例子是在模拟飞秒脉冲的载波冲击时数值色散的影响。可以为这些效应提供基准的合适模型是可积离散非线性薛定谔方程。该模型的可积性使人们能够对该系统的连续统极限进行精确的分析。PI建议使用该模型的基础Kahler几何来推断凸性估计，这应该使人们能够推导出这些极限的有效表征，包括调制不稳定性的出现。然后将这些表征与由形式平均导出的调制方程以及非线性薛定谔（NLS）方程的不可积离散化的数值模拟进行比较。第二个主题涉及非线性光学中的模式形成。实验揭示了非线性光学腔中相干场的丰富多样的横向图样。亚利桑那数学科学中心的非线性光学小组已经推导出了一个由线性极化输入场驱动的、充满各向同性非线性克尔介质的腔体的平均场模型。该模型的方程具有阻尼、驱动耦合对离焦NLS方程的形式，并已在数值上证明可以产生实验中看到的类型的图案和缺陷。PI和他的合作者特别感兴趣的是了解图案形成演化方程中缺陷的形成和演化，并开发了阶参数方程，该方程模拟了各种此类演化方程中调制滚动图案的行为。PI提出对上述平均场模型推导和研究这种调制方程。PI还开发了一种基于几何奇点理论的通用方法，用于描述pde系统中的奇点。本项目的目标是首先，对非线性光学模型的一般缺陷类型进行分类并评估其时间稳定性；第二，控制缺陷的形成，直至消除缺陷或利用缺陷在模式中编码信息。最近在一维模型中控制模式不规则性方面的成功支持了对第二个目标的期望。非线性光学，可以定义为研究强光与物质相互作用的学科，随着20世纪60年代激光的发明而诞生。在早期，这种相互作用需要非常大的强度；然而，现在这样的系统可以构建只有适度的功率要求。与此同时，也有可能制造出所谓的大孔径器件，如半导体激光器。这种器件将在许多技术领域发挥重要作用，例如具有超快交换能力的卫星通信系统。随着这些更大的物理尺度的引入，人们期望看到图案的形成以及这些图案中的缺陷（类似于晶体缺陷），这些缺陷可以通过本提案中开发的方程来建模。了解这些模式是如何产生和演变的，可以在有效地操作和控制这些设备方面发挥重要作用。在本建议的第二个项目中，将研究一类特殊的相互作用晶格系统的振荡解。这些系统具有从理论生物学到用于数值模拟光学系统的数值方法的建模应用。目标是严格地获得一个连续体描述，由偏微分方程给出，并在长空间和时间尺度上有效，在这些晶格系统的微观变化。这样的模型可以帮助解释微观系统，如α -螺旋蛋白分子或耦合激光阵列，如何集体行为，从而在宏观上，以便在比微观系统中原始存在的大得多的尺度上对现象做出贡献。* * *

9626306 Ercolani In the projects proposed here, the PI intends to concentrate on two particular areas where geometric methods are playing a fundamental role. The methods to be applied here are, first, soliton and near-integrable systems theory and, second, geometric singularity theory.The first concerns the modelling of dispersive difference schemes. The equations of nonlinear optics frequently operate in regimes where dispersive effects tend to dominate over those of viscosity. A particularly relevant example of this is the effects of numerical dispersion in modelling carrier wave shocking of femtosecond pulses. An appropriate model that can provide a benchmark for these effects is the integrable discrete nonlinear Schrodinger equations. The integrability of this model enables one to entertain a precise analysis of continuum limits for this system. The PI proposes to use the underlying Kahler geometry of this model to deduce convexity estimates which should enable one to derive a valid characterization of these limits including the appearance of modulational instabilities. These characterizations will then be compared to modulation equations derived by formal averaging as well as to numerical simulations of non-integrable discretizations of the nonlinear Schrodinger (NLS) pde. The second topic concerns pattern formation in nonlinear optics. Experiments reveal a rich variety of transverse patterns for coherent fields in a nonlinear optical cavity. The Nonlinear Optics group of the Arizona Center for Mathematical Sciences has derived mean field models for a cavity filled with an isotropic, nonlinear Kerr medium, and driven by a linearly polarised input field. The equations for this model have the form of a damped, driven coupled pair of defocussing NLS equations and have been numerically demonstrated to produce patterns and defects of the type seen experimentally. The PI and his collaborators are particularly interested to understand the formation and evolut ion of defects in pattern forming evolution equations and have developed order parameter equations which model the behaviour of modulated roll paterns in a variety of such evolution equations. The PI proposes to derive and study such modulation equations for the above mentioned mean field model. The PI has also developed a general approach, based on geometric singularity theory, for the description of singularties in systems of pde's. The goals of this project will be first, to classify generic types of defects for the nonlinear optics model and assess their temporal stability; second, to control the formation of defects to the end of either eliminating them or using them to encode information in patterns. Expectations concerning this second goal are supportied by recent sucess in controlling pattern irregularities in one dimensional models. %%% Nonlinear optics, which may be defined as the study of the interaction of intense light with matter, was born with the invention of the laser in the 1960's. In those early days such interactions required very large intensities; however, nowadays such systems can be constructed with only modest power requirements. Along with this has come the possibility of constructing large, so-called wide aperture, devices such as semiconductor lasers. Such devices will play an important role in many areas of technology such as satellite communications systems with an ultrafast switching capacity. With the introduction of these larger physical scales one expects to see the formation of patterns as well as defects in these patterns (analogous to crystalline defects) which can be modelled by the equations developed in this proposal. Understanding how such patterns arise and evolve can play an important role in efficiently operating and controlling these devices. In the second project of this proposal oscillatory solutions in a special class of interacting lattice systems will be investigated. These systems have modelling app lications which range from theoretical biology to the numerical methods used in numerically simulating optical systems. The goal is to obtain, rigorously, a continuum description, given by partial differential equations and valid on long spatial and temporal scales, of the microscopic variations in these lattice systems. Such models can help to explain how a microscopic system, such as an alpha-helix protein molecule or a coupled laser array, can behave collectively, and hence macroscopically, in order to contribute to phenomena on scales much larger than those orignally present in the microscopic system. ***