Nearly Integrable PDES: Open Mathematical Problems and Their Technological Applications

森林调制理论和对标量和矢量非线性薛定谔方程小色散极限的严格分析是非线性光纤中数据传输的基础。对于目前使用的脉冲，即所谓的“不归零”或NRZ脉冲，基本问题包括：脉冲如何退化？光纤特性和脉冲特性的什么组合决定了脉冲衰减的程度和位置？首席研究员和肯·麦克劳克林（Ken McLaughlin）针对目前使用的两种特殊的跨洋纤维研究了这些问题，并给出了上述问题的答案。提出的新研究涉及描述一旦退化开始的脉冲传播；在数学上，这些结果需要在两个基本方面推广可积偏微分方程方法。首先，必须提供激波形成后溶液的详细行为，包括波纹随主脉冲传播和演变的速度；其次，对于耦合NLS方程在存在双折射效应时的弱色散行为，迄今为止还没有结果，并且没有系统地构建耦合NLS方程的振荡解。为了实现各种计划应用的鲁棒、可靠的通信链路，必须解决当前非线性光纤中的数据传输问题。这些问题涉及到：人们使用的纤维类型；对于给定的光纤，作为信息位载体发送的数据类型；对于给定的光纤和数据，传输线在比特被扭曲而发生不可接受的错误之前能保持多长时间？对于应用数学界来说，值得注意的是，这些问题中的每一个都可以用称为近可积的特殊非线性偏微分方程的前沿问题来构建。众所周知，脉冲在现代非线性光纤中的传播是用摄动非线性薛定谔方程的解来近似的，该方程是由首席研究员花了20年时间详细研究的特殊方程。纤维的材料特性为方程提供了系数，纤维的长度决定了需要多长时间来构建解，而沿着纤维发送的数据的形状是解的输入。脉冲传播和衰减的建模涉及两个阶段：对这些特殊类型输入数据的方程解的数学理解和描述；以及将数学见解和工具转化为实际理解、问题识别和补救建议。脉冲如何退化以及如何形成脉冲以最小化退化的问题是构成本研究核心的基本数学问题。私家侦探已经并将继续与科技界进行沟通，包括到目前为止与朗讯科技公司和康宁公司的科学家进行访问和对话。

9704549 Forest Modulation theory and rigorous analysis of the small dispersion limit of scalar and vector nonlinear Schrodinger (NLS) equations are fundamental to data transmission in nonlinear fiber optics. For pulses in use today, so- called "non-return-to-zero" or NRZ pulses, fundamental issues consist of: how do the pulses degrade?, and, what combination of fiber properties and pulse features determine how much and where the pulse degrades? The principal investigator and Ken McLaughlin have studied these questions for two particular transoceanic fibers in use today, with answers provided to the above questions. Proposed new research involves the description of pulse propagation once degradation has begun; mathematically, these results require generalization of integrable PDE methods in two fundamental ways. First, one must provide detailed behavior of the solution after shock formation, including how fast the ripples spread and evolve with the main pulse; and second, there are no results to date for weak dispersive behavior of the coupled NLS equations which apply when there are birefringent effects, and there is no systematic construction of oscillatory solutions for coupled NLS pdes. Current data transmission questions in nonlinear fiber optics have to be answered in order to achieve a robust, faithful communication link for the various planned applications. These questions relate to: the type of fiber one uses; for a given fiber, the type of data one sends as carriers of information bits; for given fiber and data, how long can the transmission line be before unacceptable errors occur as bits become distorted. Remarkably for the applied mathematical community, each of these issues can be framed in terms of cutting-edge questions in special nonlinear partial differential equations called nearly integrable. The pulse propagation in modern nonlinear fibers is well-known to be approximated by the solution of perturbed nonlinear Schrodinger equations, special equations studied in great detail by the principal investigator for twenty years. The material properties of the fiber give coefficients in the equations, the length of the fiber dictates how long one must construct the solution, and the shape of the data that is sent along the fibers is the input for the solution. The modeling of pulse propagation and degradation involves two phases: the mathematical understanding and description of the solutions of these equations for these special types of input data; and the transfer of the mathematical insights and tools to practical understanding, identification of problems, and suggestions for remedies. The issues of how pulses degrade and how to shape pulses to minimize degradation are fundamental mathematical issues which form the core of the proposed research. The P.I. has and will continue to communicate with the technological community, consisting thus far of visits and conversations with staff scientists at Lucent Technologies and Corning, Inc.