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

完全可积非线性偏微分方程的逆散射变换和非衰减解

• 批准号: 1009673
• 负责人: Alexei Rybkin
• 金额: $ 20万
• 依托单位: University of Alaska Fairbanks Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1009673&HistoricalAwards=false
• 关键词: Inverse; Scattering; Transform; non; decaying

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.

该项目继续扩展求解可积非线性演化 PDE（偏微分方程）的逆散射变换 (IST) 方法，以研究更大类别的初始数据。换句话说，该项目重点研究初始剖面的孤子理论。众所周知，无穷远处的缓慢衰减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，从而揭示局部之间非常复杂的关系。偏微分方程或数值分析的标准技术无法有效解决此类问题，因为它结合了初始数据的全局和局部属性，线性化了问题，并提供了准确的渐近行为。尽管基础微分算子的谱（例如 Korteweg-de Vries (KdV) 方程中的薛定谔算子）比经典 IST 复杂得多，但它可以适当地表示为Titchmarsh-Weyl m 函数对于几乎任何合理的初始轮廓都有明确的定义，该项目的主要目的是研究这种情况下的 IST，特别是 Lax 中微分算子的不同谱分量的影响。我们将研究相应的非线性偏微分方程的解的逆散射变换，该变换最早是在 60 年代针对浅水波的 KdV 方程发现的，不久之后，它也被发现用于许多其他重要的非线性偏微分方程。被认为是数学上的根本性突破，连接了纯数学和理论物理学的不同分支，具有从流体动力学和非线性光学到天体物理学和基本粒子理论的众多应用。扩展 IST 有效性范围的重要性，这是 IST 的原理。该项目得到了数学家和物理学家的认可，其结果预计具有很强的应用性，可用于研究不同背景（包括噪声）、潮汐波、某些背景下的波传播。气象现象、理解反常波或任何其他初始数据在无穷远不接近于零的应用问题，以及流体动力学、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。鉴于其卓越的血统，其多样性该项目的主题涉及数学，应用丰富，通过研究为参与的本科生和研究生提供了良好的教育体验。