Nonlinear Geometric Optics

数学科学：非线性几何光学[0104096 . rauch]该项目涉及研究非线性双曲型偏微分方程短波解的新渐近方法的发展和分析。短波长的渐近性，也被称为几何光学，是分析偏微分方程最具穿透力和广泛应用的方法之一。该方法传统上应用于线性问题，最近被用于在非线性环境中提供严格的结果。这个项目继续发展处理非线性问题的方法。特别强调了对超短激光脉冲模型的解的分析，以及波列和脉冲在穿过焦点时的行为。这个项目开发了新的数学工具来描述波的传播，其形式是短脉冲。这种被称为渐近分析的技术，是在脉冲波长越来越短的极限下研究这种脉冲。在这种极限情况下，可以学到很多关于基本方程的解，这些方程表达了物理定律，如牛顿流体第二定律或电磁学定律。这个项目发展了分析在各种应用中出现的重要非线性方程的技术。特别注意了当几何光学射线长时间保持在一起时发生的衍射效应，以及当脉冲聚焦在非线性状态时发生的效应。有希望的初步结果表明，标准数值技术可以提高精度和简单性。该项目的结果将为重要物理系统的数值模拟提供可靠的程序，包括产生超短脉冲的激光器。

Mathematical Sciences: Nonlinear Geometric Optics0104096RauchThe project concerns the development and analysis of new asymptotic methods to study short wavelength solutions of nonlinear hyperbolic partial differential equations. Short wavelength asymptotics, also known as geometric optics, is one of the most penetrating and widely applicable methods for analyzing partial differential equations. Traditionally applied to linear problems, the method has recently been employed to furnish rigorous results in the nonlinear context. This project continues the development of the method for treatment of nonlinear problems. Special emphasis is placed on analysis of solutions that model ultrashort laser pulses and on the behavior of both wave trains and pulses upon crossing focal points.This project develops new mathematical tools to describe the propagation of waves whose form is that of short pulses. The technique, called asymptotic analysis, studies such pulses in the limit as the wavelength of the pulse becomes shorter and shorter. In such limiting situations, much can be learned about solutions of the underlying equations, which express physical laws such as Newton's second law for fluids or the laws of electromagnetism. This project develops the technique to analyze important nonlinear equations that arise in a variety of applications. Special attention is given to diffractive effects that occur when rays of geometric optics stay close together for long times, and to the effects that occur when pulses focus in a nonlinear regime. Promising preliminary results show that standard numerical techniques can be improved in both accuracy and simplicity. The results of this project will yield reliable procedures for numerical simulations of important physical systems, including lasers that produce ultrashort pulses.