Nearly Integrable PDES: Open Mathematical Problems and Their Technological Applications

小行星9704549 调制理论和严格分析的小色散极限 标量和矢量非线性薛定谔（NLS）方程是 非线性光纤中的数据传输 对于当今使用的脉冲，所谓的“不归零”或NRZ脉冲，基本问题包括： 脉冲是如何退化的？以及纤维特性和 脉冲特征决定了脉冲退化的程度和位置？ 的 首席研究员和肯·麦克劳克林研究了这些问题， 两种特殊的跨洋纤维在今天的使用，与答案提供给 以上问题。 拟议中的新研究涉及对脉搏的描述 传播一旦退化已经开始;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，必须提供冲击后溶液的详细行为 形成，包括如何快速的涟漪传播和演变的主要 脉冲;第二，到目前为止还没有弱色散行为的结果 耦合NLS方程适用时，有双折射 影响，并且没有系统的振荡解决方案的建设 耦合NLS PDE。 目前非线性光纤中的数据传输问题， 回答，以实现一个强大的，忠实的通信链路， 各种计划的应用。 这些问题涉及：纤维的类型 一个使用;对于给定的光纤，一个发送的数据类型作为 信息位;对于给定的光纤和数据， 在不可接受的错误发生之前，因为比特变得失真。 值得注意的是，对于应用数学界来说， 可以在特殊的非线性部分的前沿问题的框架 微分方程称为近可积。 脉冲在 众所周知，现代非线性光纤可以近似为 的扰动非线性薛定谔方程，特殊的方程研究， 20年来主要研究者的详细研究。 材料 纤维的性质给出方程中的系数， 纤维决定了必须构建溶液的长度，以及溶液的形状。 沿着光纤发送的数据是解决方案的输入。 建模 的脉冲传播和退化涉及两个阶段： 数学理解和描述这些解决方案 这些特殊类型的输入数据的方程;以及 数学的见解和工具，以实际的理解，确定 问题和补救措施的建议。 脉冲如何退化的问题 以及如何对脉冲进行整形以使退化最小化， 数学问题，形成拟议的研究的核心。 私家侦探 已经并将继续与技术界沟通， 包括到目前为止的访问和与朗讯的科学家交谈 技术和康宁公司

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.