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

本项目继续研究扩展的逆散射变换（IST）方法解决可积非线性演化偏微分方程（偏微分方程）处理初始数据在一个更大的类。换句话说，该项目的重点是孤子理论的初始配置文件是比快速衰减或周期性更广泛。众所周知，无穷远处的缓慢衰变可能会导致新的现象。例如，某些平滑但缓慢衰减的初始数据可能会变得粗糙甚至爆炸，揭示了局部和全局行为之间非常复杂的关系。偏微分方程或数值分析的标准技术是无效的，以解决这些问题。 IST更适合于研究这种现象，因为它结合了初始数据的全局和局部特性，线性化了问题，并提供了准确的渐近行为。虽然基础微分算子的谱（例如薛定谔在Korteweg-de弗里斯（KdV）方程的情况下）比经典IST复杂得多，但它可以用蒂奇马什-魏尔m函数来适当地表示，该函数几乎对任何合理的初始轮廓都有良好的定义。 这个项目的主旨是在这种情况下的IST的研究。特别是，不同的谱分量的微分算子的Lax对相应的非线性偏微分方程的解决方案的效果将进行调查。逆散射变换最早是在60年代由浅水波的KdV方程发现的。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，现在被认为是数学的根本突破，连接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的许多应用。扩展IST的有效性范围的重要性，这是该项目的主要目标，是公认的数学家和物理学家。所得结果具有很强的应用性，可用于研究不同背景下的波传播（包括噪声），潮汐波，某些气象现象，理解反常波或任何其他应用问题，其中初始数据在无穷大不接近零，以及在诸如流体力学，电信，大气科学，非线性光学，等离子体，天体物理学，鉴于其显着的血统，涉及数学的多样性和丰富的应用，该项目的主题提供了一个伟大的教育经验，通过研究的本科生和研究生参与。

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.