Inverse Scattering and Partial Differential Equations

逆散射和偏微分方程

基本信息

  • 批准号:
    1208778
  • 负责人:
  • 金额:
    $ 19.18万
  • 依托单位:
  • 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

项目摘要

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.
主要研究员将继续他的研究在几何分析的三个领域:(1)完全可积,在两个空间和一个时间维度的非线性色散方程,(2)混沌散射共振,和(3)逆散射薛定谔和狄拉克型方程线上奇异的潜力。2 + 1维完全可积色散方程描述了非线性表面波,表现出“块状”和线孤子行为,并提供了具有Kähler目标的薛定谔映射的例子。主要研究者和合作者将开发的逆散射和规范变换到严格的分析方法的技术完全可积色散方程的研究在2+1维,使用功能,谐波和全球分析的工具。他们寻求一个完整的图片的轨道结构和稳定性,这些动力系统的存在,分类和稳定的孤立子解决方案和汉密尔顿结构的流量。最初,PI将集中在Davey-Stewartson,Novikov-Veselov,Kadomtsev-Petviashvili和Ishimori方程作为测试用例,其动力学行为可以深入研究。渐近双曲(AH)和复双曲(CH)流形是变曲率流形,在无穷远处具有简单的几何和混沌测地线流,其特征在于一个紧凑的分形陷阱集。因此,它们是研究混沌散射的自然目标,以研究被困的测地线集和共振分布之间的关系。开发近似(AH)和精确(CH)迹线公式的工具,主要研究者和合作者将根据动态数据获得共振分布的估计。继续他的工作,对线上的奇异电位逆散射,首席研究员将研究波算子,m-功能,并逆散射薛定谔方程和狄拉克系统的地图。本工作的目标将是开发西蒙的A-函数的类似物,并研究具有奇异初始数据的NLS和KdV方程的解的定性行为。完全可积动力系统和混沌动力系统是无穷维动力系统的重要“极值”情形,它们出现在应用科学的许多不同领域。一维完全可积方法(由散射数据确定的Riemann-Hilbert问题的解)为可积偏微分方程、随机矩阵集合、圆和直线上的正交多项式以及组合问题的解提供了非常精确的渐近性。PI旨在为确定二维完全可积偏微分方程解的振荡d杆问题开发类似的渐近方法,平面正交多项式的渐近性和正态矩阵分布的渐近性。这种分析将需要新的技术和谐波分析的结果。在另一个极端,混沌动力系统的量子化是当前研究兴趣的一个领域:在拟议的研究中,我们将研究经典捕获和量子混沌之间的关系,在几何设置中的动力学和散射是经得起详细分析。

项目成果

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Peter Perry其他文献

The Management of University-Industry Collaborations Involving Empirical Studies of Software Enginee
涉及软件工程实证研究的产学合作管理
Isoscattering deformations for complete manifolds of negative curvature
  • DOI:
    10.1007/bf02922135
  • 发表时间:
    2006-12-01
  • 期刊:
  • 影响因子:
    1.500
  • 作者:
    Peter Perry;Dorothee Schueth
  • 通讯作者:
    Dorothee Schueth
Some examples inL p spectral geometry
  • DOI:
    10.1007/bf02921315
  • 发表时间:
    1993-07-01
  • 期刊:
  • 影响因子:
    1.500
  • 作者:
    Robert Brooks;Peter Perry;Peter Petersen V
  • 通讯作者:
    Peter Petersen V
Pastoralism and politics: reinterpreting contests for territory in Auckland Province, New Zealand, 1853–1864
  • DOI:
    10.1016/j.jhg.2007.10.001
  • 发表时间:
    2008-04-01
  • 期刊:
  • 影响因子:
  • 作者:
    Vaughan Wood;Tom Brooking;Peter Perry
  • 通讯作者:
    Peter Perry
Closed Geodesics in Homology Classes for Convex Co-Compact Hyperbolic Manifolds
  • DOI:
    10.1023/a:1016226616826
  • 发表时间:
    2002-01-01
  • 期刊:
  • 影响因子:
    0.500
  • 作者:
    Jeffrey McGowan;Peter Perry
  • 通讯作者:
    Peter Perry

Peter Perry的其他文献

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{{ truncateString('Peter Perry', 18)}}的其他基金

Conference and Workshop: Scattering and Inverse-Scattering in Multi-Dimensions, May 16-23, 2014
会议和研讨会:多维散射和逆散射,2014 年 5 月 16-23 日
  • 批准号:
    1408891
  • 财政年份:
    2014
  • 资助金额:
    $ 19.18万
  • 项目类别:
    Standard Grant
CBMS Regional Conference in the Mathematical Sciences - Global Harmonic Analysis - June 2011
CBMS 数学科学区域会议 - 全球调和分析 - 2011 年 6 月
  • 批准号:
    1040927
  • 财政年份:
    2011
  • 资助金额:
    $ 19.18万
  • 项目类别:
    Standard Grant
Spectral Problems in Geometry and Partial Differential Equations
几何和偏微分方程中的谱问题
  • 批准号:
    0710477
  • 财政年份:
    2007
  • 资助金额:
    $ 19.18万
  • 项目类别:
    Standard Grant
Inverse Problems in Geometry and Partial Differential Equations
几何反问题和偏微分方程
  • 批准号:
    0408419
  • 财政年份:
    2004
  • 资助金额:
    $ 19.18万
  • 项目类别:
    Standard Grant
Conference on Inverse Spectral Geometry, June 20-28, 2002, Lexington, Kentucky
逆谱几何会议,2002 年 6 月 20-28 日,肯塔基州列克星敦
  • 批准号:
    0207125
  • 财政年份:
    2002
  • 资助金额:
    $ 19.18万
  • 项目类别:
    Standard Grant
Spectral Geometry of Non-Compact Domains and Riemannian Manifolds
非紧域和黎曼流形的谱几何
  • 批准号:
    0100829
  • 财政年份:
    2001
  • 资助金额:
    $ 19.18万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Spectral Geometry of Compact Riemannian Manifolds and Kleinian Groups
数学科学:紧致黎曼流形和克莱因群的谱几何
  • 批准号:
    9707051
  • 财政年份:
    1997
  • 资助金额:
    $ 19.18万
  • 项目类别:
    Standard Grant
High-Performance Computing Laboratory
高性能计算实验室
  • 批准号:
    9508543
  • 财政年份:
    1995
  • 资助金额:
    $ 19.18万
  • 项目类别:
    Standard Grant
Mathematical Sciences: Research Experiences for Undergraduates - Inverse Problems: Mathematics and Engineering
数学科学:本科生研究经历-反问题:数学与工程
  • 批准号:
    9424012
  • 财政年份:
    1995
  • 资助金额:
    $ 19.18万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: Spectral Geometry and Inverse Problems
数学科学:谱几何和反问题
  • 批准号:
    9203529
  • 财政年份:
    1992
  • 资助金额:
    $ 19.18万
  • 项目类别:
    Continuing Grant

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