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

本课题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，以开发一种反光谱变换，将该方法的有效性范围扩展到包括远程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。规划了计算模拟来指导分析。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解的方法，以适应更现实的初始数据类别。这将通过将两个杰出的理论——孤子理论和蒂奇沼泽-魏尔理论——联系在一起来实现。孤子理论起源于对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想不到的联系的惊人发现。它被认为是数学的一个根本性突破，连接了纯数学和理论物理的不同分支，从流体力学和非线性光学到天体物理学和基本粒子理论都有许多应用。Titchmarsh-Weyl理论是在Sturm-Liouville问题的基础上发展起来的，它已成为常微分算子谱分析的基石。本项目致力于开发一种利用Titchmarsh-Weyl理论的孤子理论方法。这项工作将为可积非线性演化系统提供更好的数学理解，这可以促进流体力学、光纤通信和等离子体理论中非线性波研究的新发展。

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.