Titchmarsh - Weyl m-function and integrable nonlinear partial differential equations

Titchmarsh - Weyl m 函数和可积非线性偏微分方程

• 批准号: 0707476
• 负责人: Alexei Rybkin
• 金额: --
• 依托单位: University of Alaska Fairbanks Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2010-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707476&HistoricalAwards=false
• 关键词: Titchmarsh; Weyl; function; integrable; nonlinear

This project investigates an extension of the inverse scattering transform (IST) method of solution for integrable nonlinear partial differential equations to handle initial data in a larger class. The scattering data used in connection with short-range potentials will be replaced by data involving the Titchmarsh-Weyl m-function to develop an inverse spectral transform that extends the range of validity of the method to initial conditions that include long-range and oscillatory functions. The work will establish a properly regularized Marchenko equation that permits reconstruction of the potential. Computational simulations to guide the analysis are planned.The main goal of the proposed research is to extend methods for explicit solution of certain nonlinear evolution equations to accommodate more realistic classes of initial data. This will be achieved by linking together two remarkable theories, Soliton Theory and Titchmarsh-Weyl Theory. Soliton theory originated in the striking discovery of an unexpected link between the Korteweg-de Vries equation in nonlinear hydrodynamics and quantum scattering theory. It is regarded as a fundamental breakthrough in mathematics, connecting different branches of pure mathematics and theoretical physics, with numerous applications ranging from hydrodynamics and nonlinear optics to astrophysics and elementary particle theory. Titchmarsh-Weyl theory was developed in the connection with the Sturm-Liouville problem, which has become a cornerstone of the spectral analysis of ordinary differential operators. This project is devoted to developing an approach to soliton theory that takes advantage of Titchmarsh-Weyl theory. The work will provide improved mathematical understanding of integrable nonlinear evolution systems, which can stimulate new developments in the study of nonlinear waves in hydrodynamics, fiber optics communication, and plasma theory.

该项目研究可积非线性偏微分方程解的逆散射变换 (IST) 方法的扩展，以处理更大类别的初始数据。 与短程势相关的散射数据将被涉及 Titchmarsh-Weyl m 函数的数据所取代，以开发逆谱变换，将方法的有效性范围扩展到包括长程函数和振荡函数的初始条件。 这项工作将建立一个适当正则化的马尔琴科方程，允许重建势能。 计划进行计算模拟来指导分析。所提出的研究的主要目标是扩展某些非线性演化方程的显式求解方法，以适应更现实的初始数据类别。 这将通过将孤子理论和蒂奇马什-韦尔理论这两个著名理论联系在一起来实现。 孤子理论起源于非线性流体动力学中的 Korteweg-de Vries 方程与量子散射理论之间意外联系的惊人发现。 它被认为是数学上的根本性突破，连接了纯数学和理论物理学的不同分支，具有从流体动力学和非线性光学到天体物理学和基本粒子理论的众多应用。 Titchmarsh-Weyl 理论是结合 Sturm-Liouville 问题而发展起来的，该问题已成为普通微分算子谱分析的基石。 该项目致力于开发一种利用 Titchmarsh-Weyl 理论的孤子理论方法。 这项工作将提高对可积非线性演化系统的数学理解，从而刺激流体动力学、光纤通信和等离子体理论中非线性波研究的新发展。