Asymptotic Problems in Nonlinear Waves and Beyond

本项目涉及可积非线性波动方程渐近分析工具的开发。这项工作有两个互补的方面：线性微分方程散射理论的渐近分析和相应逆问题的渐近分析。沿路研究的具体应用问题包括：(i)聚焦非线性薛定谔方程的半经典极限，（ii）修正非线性薛定谔方程的相应半经典极限，（iii）实验室坐标下正弦-戈登方程的半经典极限，（iv） Ablowitz-Ladik方程的连续统极限，(v)多分量可积方程的奇异极限。(6)逼近理论中的几个渐近问题及其在随机矩阵理论中的应用。这项工作是有趣和重要的，因为它将促进对奇异极限的理解，导致正式的不适定动力系统。事实上，具有一般光滑但非解析初始数据的聚焦非线性薛定谔方程的半经典极限的激励问题仍然是应用分析中最重要的开放问题之一，并且作为该项目的一部分开发的工具将直接解决该问题和其他类似问题。此外，虽然该项目的基本目标是发展分析方法，但这些方法也将应用于几个特定的可积方程以及非线性波领域以外的开放问题。例如，我们打算将在非线性波动理论背景下开发的专门的渐近方法应用于正态随机矩阵模型的相关函数的渐近分析（在体积缩放极限下）问题，并将其与拉普拉斯增长理论（也称为Hele-Shaw流）和保角映射的一致联系。作为这个项目的一部分，许多问题都有一个普遍的特征，在各种物理现象的建模中产生，因此，适用于这些问题的分析技术具有深远的影响。这个项目也有教育方面的内容，强调通过合作研究和课程开发来培养博士后和研究生。

This project concerns the development of tools for the asymptotic analysis of integrable nonlinear wave equations. There are two complementary aspects of this work: asymptotic analysis in the scattering theory of linear differential equations, and asymptotic analysis of the corresponding inverse problems. The particular application problems to be studied along the way include (i) the semiclassical limit of the focusing nonlinear Schrodinger equation, (ii) the corresponding semiclassical limit of the modified nonlinear Schrodinger equation, (iii) the semiclassical limit of the sine-Gordon equation in laboratory coordinates, (iv) the continuum limit of the Ablowitz-Ladik equations, (v) singular limits for multicomponent integrable equations, and (vi) several asymptotic problems in approximation theory with applications to random matrix theory.This work is interesting and important because it will promote understanding of singular limits leading formally to ill-posed dynamical systems. Indeed, the motivating problem of the semiclassical limit of the focusing nonlinear Schrodinger equation with general smooth but nonanalytic initial data remains one of the most important open problems in applied analysis, and the tools developed as part of this project will directly address this problem and other similar ones. Furthermore, while the basic aim of the project is the development of methods of analysis, the methods will also be applied to several specific integrable equations and also to open problems beyond the field of nonlinear waves. For example, we intend to apply specialized asymptotic methods developed in the context of nonlinear wave theory to the problem of the asymptotic analysis (in the bulk scaling limit) of the correlation functions of the normal random matrix model, with its coincident connections to the theory of Laplacian growth (also known as Hele-Shaw flow) and conformal mapping. Many of the problems addressed as part of this project have a universal character, arising in the modeling of diverse physical phenomena, and it follows that analytical techniques applicable to these problems have far-reaching consequences. This project also has an educational component, stressing the training of postdocs and graduate students through collaborative research and course development.