Inverse Scattering and Partial Differential Equations

逆散射和偏微分方程

• 批准号: 1208778
• 负责人: Peter Perry
• 金额: $ 19.18万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1208778&HistoricalAwards=false
• 关键词: Inverse; Scattering; Partial; Differential; Equations

The Principle Investigator will continue his research in three areas of geometric analysis: (1) Completely integrable, nonlinear dispersive equations in two space and one time dimensions, (2) resonances in chaotic scattering, and (3) inverse scattering for Schrödinger and Dirac-type equations on the line with singular potentials. Completely integrable dispersive equations in 2 + 1 dimensions describe nonlinear surface waves, exhibit "lump" and line-soliton behavior, and provide examples of Schrödinger maps with Kähler targets. The Principal Investigator and collaborators will develop the techniques of inverse scattering and gauge transformations into rigorous analytical methods for the study of completely integrable dispersive equations in 2+1 dimensions, using the tools of functional, harmonic, and global analysis. They seek a complete picture of the orbit structure and stability of these dynamical systems; existence, classification, and stability of soliton solutions; and Hamiltonian structure of the flows. Initially the PI will concentrate on the Davey-Stewartson, Novikov-Veselov, Kadomtsev-Petviashvili and Ishimori equations as test cases whose dynamical behavior can be studied in depth. Asymptotically hyperbolic (AH) and complex hyperbolic (CH) manifolds are variable-curvature manifolds with simple geometry at infinity and chaotic geodesic flows characterized by a compact, fractal trapped set. As such they are natural targets for the investigation of chaotic scattering, to study the relationship between the trapped set of geodesics and the distribution of resonances. Developing the tools of approximate (AH) and exact (CH) trace formulae, the Principal Investigator and collaborators will obtain estimates on the distribution of resonances in terms of dynamical data. Continuing his work on inverse scattering for singular potentials on the line, the Principal Investigator will study wave operators, m-functions, and inverse scattering maps for Schrödinger equations and Dirac systems. The goal of this work will be to develop analogues of Simon's A-function and to study qualitative behavior of solutions to the NLS and KdV equations with singular initial data. Completely integrable and chaotic dynamical systems are important "extremal" cases of infinite-dimensional dynamical systems which occur in many different areas of applied science. The completely integrable method in one dimension (formulated as the solution of a Riemann-Hilbert problem determined by scattering data) gives remarkably precise asymptotics for solutions of integrable partial differential equations, random matrix ensembles, orthogonal polynomials on the circle and the line, and combinatorial problems. The PI seeks to develop analogous asymptotic methods for the oscillatory d-bar-problems that determine solutions of completely integrable partial differential equations in two dimensions, asymptotics of orthogonal polynomials in the plane, and asymptotics of normal matrix distributions. This analysis will require new techniques and results in harmonic analysis. At the other extreme, the quantization of chaotic dynamical systems is an area of intensive current research interest: in the proposed research we will study the relationship between classical trapping and quantum chaos in a geometrical setting where the dynamics and scattering are amenable to a detailed analysis.

首席研究员将继续在几何分析的三个领域进行研究：（1）两个空间和一个时间维度中的完全可积非线性色散方程，（2）混沌散射中的共振，以及（3）薛定谔和狄拉克的逆散射具有奇异势的直线方程描述了非线性表面波，表现出“团”和线孤子行为，并提供了 2 + 1 维的完全可积色散方程。首席研究员和合作者将使用泛函、调和和数学工具，将逆散射和规范变换技术发展为严格的分析方法，用于研究 2+1 维中的完全可积色散方程。全局分析。他们寻求这些动力系统的轨道结构和稳定性；孤子解的存在、分类和稳定性；最初，PI 将集中于流的哈密顿结构。 Davey-Stewartson、Novikov-Veselov、Kadomtsev-Petviashvili 和 Ishimori 方程作为测试用例，可以深入研究其动力学行为。渐近双曲 (AH) 和复双曲 (CH) 流形是具有无穷大和混沌的简单几何的变曲率流形。测地流以紧凑的分形俘获集为特征，因此它们是研究混沌散射的自然目标。研究测地线捕获集与共振分布之间的关系 通过开发近似 (AH) 和精确 (CH) 迹公式工具，首席研究员和合作者将根据动态数据获得共振分布的估计。首席研究员将继续研究线上奇异势的逆散射工作，研究薛定谔方程和狄拉克系统的波算子、m 函数和逆散射图。开发西蒙 A 函数的类似物，并研究具有奇异初始数据的 NLS 和 KdV 方程解的定性行为，完全可积和混沌动力系统是无限维动力系统的重要“极端”情况，发生在许多不同的领域。一维完全可积方法（由散射数据确定的黎曼-希尔伯特问题的解）给出了可积偏微分解的精确渐近性。方程、随机矩阵系综、圆和直线上的正交多项式以及组合问题 PI 致力于为振荡 d 条问题开发类似的渐近方法，以确定二维完全可积偏微分方程的解，以及组合问题。平面中的正交多项式和正态矩阵分布的渐近性将需要调和分析中的新技术和结果。在另一个极端，混沌动力系统的量子化是当前研究热点的一个领域：在拟议的研究中，我们将在动力学和散射服从于几何设置的情况下研究经典捕获和量子混沌之间的关系。详细分析。