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

该项目研究了求解可积非线性偏微分方程组的逆散射变换(IST)方法的扩展，以处理更大类中的初始数据。与短程势有关的散射数据将被涉及Titchmarsh-Weyl m函数的数据所取代，以开发将该方法的有效范围扩展到包括长程函数和振荡函数的初始条件的逆谱变换。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。研究的主要目的是扩展某些非线性发展方程显式解的方法，以适应更真实的初始数据类。这将通过将两个引人注目的理论--孤子理论和蒂奇马什-韦尔理论--联系在一起来实现。孤子理论起源于一个惊人的发现，发现了非线性流体力学中的Korteweg-de Vries方程和量子散射理论之间的一种意想不到的联系。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的不同分支，有着从流体力学和非线性光学到天体物理和基本粒子理论的众多应用。Titchmarsh-Weyl理论是在Sturm-Liouville问题的基础上发展起来的，Sturm-Liouville问题已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Titchmarsh-Weyl理论的孤子理论的方法。这项工作将提高对可积非线性演化系统的数学理解，促进流体力学、光纤通信和等离子体理论中的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.