Inverse Scattering Transform and non-decaying solutions of completely integrable nonlinear PDE's

这个项目继续研究逆散射变换(IST)方法的扩展，该方法求解可积的非线性发展偏微分方程组(PDE)，以处理更大类别的初始数据。换句话说，该项目专注于初始轮廓的孤子理论，这些轮廓比快速衰变或周期性的轮廓要广泛得多。众所周知，无限大的缓慢衰变可能会导致新的现象。例如，某些平稳但缓慢衰减的初始数据可能会变成粗糙甚至爆炸，揭示了局部行为和全局行为之间非常复杂的关系。标准的偏微分方程组或数值分析技术不能有效地解决这些问题。IST更适合于研究这类现象，因为它结合了初始数据的全局和局部性质，使问题线性化，并提供了准确的渐近行为。尽管基本微分算子的谱(例如Korteweg-de Vries(KdV)方程中的薛定谔)比经典IST的谱要复杂得多，但它可以适当地用Titchmarsh-Weyl m函数来表示，该函数对于几乎任何合理的初始轮廓都是定义良好的。本项目的主旨是对这种背景下的IST进行研究。特别地，将研究Lax对中微分算子的不同谱分量对相应的非线性偏微分方程解的影响。对于浅水波的KdV方程，60年代首次发现了逆散射变换。不久之后，它被发现用于许多其他重要的非线性偏微分方程组，现在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，从流体力学和非线性光学到天体物理和基本粒子理论，有着广泛的应用。数学家和物理学家都认识到，扩大IST的有效范围的重要性，这是该项目的主要目标。研究结果可望具有非常实用的性质，可用于研究不同背景(包括噪声)、潮汐波、某些气象现象、理解异常波或任何其他应用问题，如流体力学、电信、大气科学、非线性光学、等离子体、天体物理学等不同学科的初始数据在无穷远处不趋近于零的情况下的波传播。

This project continues to investigate an extension of the inverse scattering transform (IST) method of solving integrable nonlinear evolution PDEs (partial differential equations) to handle initial data in a larger class. In other words, the project focuses on soliton theory for initial profiles that are much broader than rapidly decaying or periodic. It is well-known that slow decay at infinity may lead to new phenomena. For instance, certain smooth but slowly decaying initial data may turn into rough or even blow-up revealing a very complicated relation between local and global behaviors. Standard techniques of PDEs or numerical analysis are ineffective to tackle such issues. The IST is much better suited to study such phenomena as it combines both global and local properties of initial data, linearizes the problem, and provides accurate asymptotic behavior. Although the spectrum of the underlying differential operators (e.g. Schrödinger in the case of the Korteweg-de Vries (KdV) equation) is much more complicated than for the classical IST, it can be suitably expressed in terms of the Titchmarsh-Weyl m-function which is well-defined for virtually any reasonable initial profile. The main thrust of this project is a study of the IST in this setting. In particular, the effect of different spectral components of the differential operator in the Lax pair on the solution of the corresponding nonlinear PDE will be investigated. The inverse scattering transform was first discovered in the 60s for the KdV equation of shallow-water waves. Soon after, it was found for many other important nonlinear PDEs and now 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. The importance of extending the range of validity of IST, which is the principal aim of the project, is recognized by both mathematicians and physicists. The results are expected to be of a very applied nature and could be employed for the study of wave propagation on different backgrounds (including noisy), tidal waves, certain meteorological phenomena, understanding freak waves or any other applied problems where initial data do not approach zero at infinity, and in such diverse disciplines as hydrodynamics, telecommunication, atmospheric sciences, nonlinear optics, plasma, astrophysics, etc. Given their remarkable pedigree, diversity of mathematics involved, and richness of applications, the topics of the project provide a great educational experience through research for the undergraduate and graduate students involved.