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.

本计画探讨可积非线性偏微分方程式之逆散射转换法之延伸，以处理更大类的初始资料。 与短程电位相关的散射数据将被涉及Titchmarsh-Weyl m函数的数据所取代，以开发一种逆谱变换，该逆谱变换将该方法的有效性范围扩展到包括长程和振荡函数的初始条件。 这项工作将建立一个适当的正规化的马尔琴科方程，允许重建的潜力。 计算模拟，以指导分析的计划。拟议的研究的主要目标是扩展方法，显式求解某些非线性演化方程，以适应更现实的初始数据类。 这将通过把两个著名的理论，孤立子理论和蒂奇马什-魏尔理论联系在一起来实现。 孤立子理论起源于一个惊人的发现，即非线性流体力学中的Korteweg-de弗里斯方程与量子散射理论之间存在着意想不到的联系。 它被认为是数学的根本突破，连接了纯数学和理论物理的不同分支，具有从流体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理论是在Sturm-Liouville问题中发展起来的，它已经成为常微分算子谱分析的基石。 这个项目致力于发展一种利用蒂奇马什-外尔理论的孤子理论的方法。 这项工作将提供更好的数学理解可积非线性演化系统，这可以刺激新的发展，在流体力学，光纤通信和等离子体理论的非线性波的研究。