Collaborative Research: Spectra of Linear Differential Operators and Turbulence in Integrable Systems

非线性演化方程，也称为孤子方程，被用来模拟各种物理系统，如海浪或光纤通信。许多这样的方程，例如浅水波浪的Korteweg-de Vries(KdV)方程，都有一个令人兴奋的特点，即它们的许多解都可以用一个显式精确地给出。这使人们能够对潜在的物理过程有一个准确的理解。然而，目前已知精确解的条件范围是受限的，这限制了此类解在现实世界应用中的使用。这个项目的目的是找到孤子方程的更大的精确解族，并使用这些解来发展相应物理系统的统计理论。该项目的主要目标是构造和研究KdV、非线性薛定谔和Kadomtsev-Petviashvili等孤子方程的新的解族。这些解是作为多孤子解的极限得到的，并且在无穷远处是有界的且非递减的。它们可以用Riemann-Hilbert问题来描述，并且可以有效地进行数值计算。第一个目标是对这些新解决方案进行严格的数学描述。PIS将调查这些解决方案在多大程度上解决了KDV和相关系统的初值问题。他们将研究相关线性算子的光谱性质，并构建非周期的一维理想导体。最后，PI将为KdV和其他孤子方程发展可积湍流的统计理论。

Nonlinear evolution equations, also known as soliton equations, are used to model a wide variety of physical systems, such as ocean waves or fiber optic communications. Many such equations, for example the Korteweg-de Vries (KdV) equation for waves in shallow water, have the exciting feature that many of their solutions can be given exactly by an explicit formula. This allows one to develop a precise understanding of the underlying physical processes. However, the range of conditions for which exact solutions are currently known is restrictive, inhibiting the use of such solutions in real-world applications. This project aims to find larger families of exact solutions to soliton equations, and use these solutions to develop statistical theories of the corresponding physical systems. The main goal of the project is to construct and study new families of solutions of soliton equations such as KdV, Nonlinear Schrödinger, and Kadomtsev-Petviashvili. These solutions are obtained as limits of multisoliton solutions, and are bounded and non-decreasing at infinity. They are described by a Riemann-Hilbert problem and can be efficiently computed numerically. The first goal is a rigorous mathematical description of these new solutions. The PIs will investigate to what extent these solutions solve the initial value problem for KdV and related systems. They will study spectral properties of the associated linear operators and construct non-periodic one-dimensional ideal conductors. Finally, the PIs will develop a statistical theory of integrable turbulence for KdV and other soliton equations.