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.

原则研究者将继续在几何分析的三个领域继续进行研究：（1）完全可以集成的非线性分散方程在两个空间和一个时间维度中，（2）混乱散射的共振，以及（3）Schrödinger和diracty type方程的逆散射与奇异的电位。在2 + 1个维度中完全可以集成的分散方程描述了非线性表面波，暴露的“肿块”和线 - 索顿行为，并提供了带有Kähler目标的Schrödinger地图的示例。首席研究者和合作者将使用功能，谐波和全局分析的工具，将反向散射和规变量转换的技术开发为严格的分析方法，以研究2+1维的完全集成的分散方程。他们寻求这些动态系统的轨道结构和稳定性的完整图片；孤子解决方案的存在，分类和稳定性；和汉密尔顿流的结构。最初，PI将集中在Davey-Stewartson，Novikov-Veselov，Kadomtsev-Petviashvili和Ishimori方程式上，作为可以深入研究动态行为的测试用例。不对称的双曲线（AH）和复杂双曲线（CH）歧管是可变曲面的歧管，在无穷大时具有简单的几何形状，而混乱的地理流动流则以紧凑的，分形的捕获套件为特征。因此，它们是混乱散射投资的自然目标，可以研究被困的大地测量学与共振分布之间的关系。开发近似（AH）和精确（CH）痕量公式的工具，主要研究者和合作者将获得有关动态数据共振分布的估计。继续他的工作在线上奇异电位的反向散射方面，首席研究员将研究波操作员，m功能和schrödinger方程和狄拉克系统的逆散射图。这项工作的目的是开发Simon A功能的类比，并使用具有单数初始数据的NLS和KDV方程的解决方案的定性行为。完全可以整合且混乱的动态系统是在应用科学的许多不同领域中发生的无限二维动态系统的重要“极端”案例。一维中完全可以整合的方法（作为通过散射数据确定的riemann-hilbert问题的解决方案）提供了非常精确的渐近渐近物，用于集成偏微分方程的溶液，随机矩阵集成，正交多种元素在圆和线上和线路上以及组合问题上。 PI试图为振荡性D-BAR问题开发类似的渐近方法，这些方法在两个维度，平面中正交多项式的渐近多项式的渐近差异和正常基质分布的渐近线分布中确定完全可以整合的偏微分方程的溶液。该分析将需要新技术并导致谐波分析。在另一个极端情况下，混乱动态系统的量化是一个密集的当前研究兴趣的领域：在拟议的研究中，我们将在几何环境中研究经典陷阱与量子混乱之间的关系，在几何环境中，动力学和散射可以通过详细的分析。