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The nonlinear Schrodinger equation, its physical origins, and the spectral measures of random matrices

The nonlinear Schrodinger equation, its physical origins, and the spectral measures of random matrices
非线性薛定谔方程、其物理起源以及随机矩阵的谱测度
批准号:
1265868
负责人:
Rowan Killip
金额:
$23.6万
依托单位国家:
美国
项目类别:
Continuing Grant
财政年份:
2013
资助国家:
美国
项目状态:
已结题
起止时间:
2013-07-01 至 2016-06-30

项目摘要

项目成果

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中文摘要
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英文摘要
The main purpose of the proposed research is to further the mathematical understanding of several standard simple physical models: the nonlinear Schrodinger equation and its connections to many-body quantum mechanics; the scattering resonances of a random black-box system; and the spectral measure of CMV matrices with random decaying coefficients. A major theme of the NLS portion of the project is to investigate the effect of obstacles. This is a concrete and challenging model case for the investigation of NLS in general geometries. Examples of dispersive equations that are scaling-critical but on the very cusp of super-criticality will also be investigated; this is an ideal proving ground for existing energy- and mass-critical techniques. Two classes of physically motivated NLS models with combined attractive and repulsive nonlinearities will be investigated. These models have already started to reveal interesting and unusual variational and dynamical structures that warrant further investigation. Additionally, the distribution of resonances associated to finite dimensional quantum systems (chosen uniformly at random, respecting time-reversal symmetry) will be investigated. Related methods will be employed to investigate the fine spectral properties for discrete Dirac equations with decaying random potentials.The theory of random matrices is driven by empirical data as surely as any physical science: Its roots lie in statistical analysis (specifically, ANOVA) and the analysis of experimental energy-level data. More recently, random matrix statistics have been observed in the vibrations of drums and the behaviour of the prime numbers. The researches of this project are aimed at helping to elucidate and explain the results of these truly mathematical experiments. Although the nonlinear Schrodinger equation is used as a simple effective model for several physical phenomena, the goal of this project is to further our understanding of the behaviour of solutions to this equation principally as a model for general evolution equations. As a step towards the vicissitudes of laboratory science/engineering, we will consider evolution in the presence of obstacles and other irregularities as well as incorporating more complicated nonlinear effects that cannot be described by a simple power law. The training of graduate students as educators and researchers in the mathematical sciences is also a significant portion of the project.
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Integrable Partial Differential Equations as Pathfinders in Mathematical Physics
The Korteweg-de Vries Equation and Beyond
  • 批准号:
    1856755
  • 项目类别:
    Continuing Grant
  • 资助金额:
    $24.86万
  • 财政年份:
    2019
  • 负责人:
    Rowan Killip
  • 依托单位:
Linear and nonlinear problems in dispersive Partial Differential Equations
  • 批准号:
    1600942
  • 项目类别:
    Continuing Grant
  • 资助金额:
    $24.0万
  • 财政年份:
    2016
  • 负责人:
    Rowan Killip
  • 依托单位:
Simple models in Mathematical Physics: Random matrices and NLS
  • 批准号:
    1001531
  • 项目类别:
    Continuing Grant
  • 资助金额:
    $25.0万
  • 财政年份:
    2010
  • 负责人:
    Rowan Killip
  • 依托单位:
国内基金
海外基金
图随机Schrodinger算子的量子噪声方法
  • 批准号:
    --
  • 项目类别:
    地区科学基金项目
  • 资助金额:
    29万元
  • 批准年份:
    2022
  • 负责人:
    王才士
  • 依托单位:
非线性Schrodinger方程耦合电磁理论的变分方法研究
  • 批准号:
    12001198
  • 项目类别:
    青年科学基金项目
  • 资助金额:
    24.0万元
  • 批准年份:
    2020
  • 负责人:
    黄文涛
  • 依托单位:
带不定位势的拟线性Schrodinger方程及相关问题
  • 批准号:
    --
  • 项目类别:
    面上项目
  • 资助金额:
    51万元
  • 批准年份:
    2020
  • 负责人:
    刘轼波
  • 依托单位:
非线性边界条件下Schrodinger方程的拟周期解
  • 批准号:
    11701212
  • 项目类别:
    青年科学基金项目
  • 资助金额:
    23.0万元
  • 批准年份:
    2017
  • 负责人:
    常晶
  • 依托单位: