Asymptotic Dynamics of Nonlinear Wave and Dispersive Equations

非线性波和色散方程的渐近动力学

基本信息

  • 批准号:
    1954707
  • 负责人:
  • 金额:
    $ 15.34万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2020
  • 资助国家:
    美国
  • 起止时间:
    2020-07-01 至 2024-06-30
  • 项目状态:
    已结题

项目摘要

Many wave propagation phenomena in the natural sciences and in engineering can be modeled by nonlinear wave and dispersive equations. While the theory of linear wave equations predicts that waves spread out and decay as time goes by, once nonlinear effects are taken into account, this changes drastically. In particular, linear and nonlinear effects may balance out to create so-called soliton solutions, whose shapes persist and refuse to disperse. It is widely believed that solutions to most nonlinear wave and dispersive equations with "generic" initial data should eventually decompose into a finite number of solitons plus a radiative term that goes to zero. This project concentrates on two themes that play an important role in the quest to understand this grand picture how waves propagate overtime. The principal investigator (PI) will develop new methods and techniques for the study of the asymptotic stability of solitons in the presence of strong nonlinear interactions. Asymptotic stability refers to the phenomenon that if a soliton gets pushed a little bit, it may wiggle for a while, but ultimately return to a form similar to the one it began with. Further, the PI will investigate the long-time dynamics of solutions to nonlinear wave equations with generic randomized initial data in several novel regimes.More specifically, the PI will use harmonic analysis and vector field techniques together with tools from spectral theory and probability theory to work toward the following goals: (1) carry out a program to obtain precise asymptotics of small solutions to one-dimensional Klein-Gordon equations with variable coefficient nonlinearities, which are related to asymptotic stability questions for "kink" solitons in numerous field theories in physics; (2) develop a modulational approach for proving the (co-dimensional) stability of certain solitons arising in some quasilinear geometric wave equations; and (3) initiate the study of the long-time dynamics of solutions to geometric wave equations and to wave equations with long-range nonlinearities for random initial data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
自然科学和工程中的许多波传播现象都可以用非线性波动和色散方程来模拟。虽然线性波动方程理论预测波随着时间的推移而传播和衰减,但一旦考虑到非线性效应,这种情况就会发生巨大的变化。具体地说,线性和非线性效应可能会平衡,从而产生所谓的孤子解,其形状持续存在,拒绝分散。人们普遍认为,具有“一般”初始数据的大多数非线性波动和色散方程的解最终应该分解为有限数量的孤子加上一个接近于零的辐射项。这个项目集中在两个主题上,这两个主题在理解这幅宏伟的图景中发挥着重要作用,波是如何超时传播的。首席研究员(PI)将发展新的方法和技术来研究存在强非线性相互作用的孤子的渐近稳定性。渐近稳定性是指,如果一个孤子受到一点推动,它可能会摆动一段时间,但最终会恢复到与它开始时类似的形式。更具体地说,PI将使用调和分析和矢量场技术,结合谱理论和概率论的工具,致力于以下目标:(1)执行一个程序来获得一维变系数非线性Klein-Gordon方程小解的精确渐近性,这与物理中许多场论中的“扭结”孤子的渐近稳定性问题有关;(2)发展一种调制方法来证明某些拟线性几何波动方程中的某些孤子的(共维)稳定性;以及(3)开始研究几何波动方程和具有长程非线性的随机初始数据波动方程的解的长期动力学。该奖项反映了NSF的法定使命,并通过使用基金会的智力优势和更广泛的影响审查标准进行评估,被认为值得支持。

项目成果

期刊论文数量(3)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
On Modified Scattering for 1D Quadratic Klein–Gordon Equations With Non-Generic Potentials
具有非泛势的一维二次克莱因-戈登方程的修正散射
Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities
  • DOI:
    10.1007/s00205-021-01675-y
  • 发表时间:
    2020-06
  • 期刊:
  • 影响因子:
    2.5
  • 作者:
    Hans Lindblad;Jonas Lührmann;A. Soffer
  • 通讯作者:
    Hans Lindblad;Jonas Lührmann;A. Soffer
Soliton dynamics for the 1D quadratic Klein-Gordon equation with symmetry
具有对称性的一维二次 Klein-Gordon 方程的孤子动力学
  • DOI:
    10.1016/j.jde.2022.10.030
  • 发表时间:
    2023
  • 期刊:
  • 影响因子:
    2.4
  • 作者:
    Li, Yongming;Lührmann, Jonas
  • 通讯作者:
    Lührmann, Jonas
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Jonas Luhrmann其他文献

Decay and asymptotics for the one-dimensional Klein-Gordon equation with variable coefficient cubic nonlinearities
具有变系数三次非线性的一维 Klein-Gordon 方程的衰变和渐近
  • DOI:
  • 发表时间:
    2020
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Hans Lindblad;Jonas Luhrmann;Avy Soffer
  • 通讯作者:
    Avy Soffer

Jonas Luhrmann的其他文献

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{{ truncateString('Jonas Luhrmann', 18)}}的其他基金

CAREER: New Frontiers in the Dynamics of Topological Solitons
职业:拓扑孤子动力学的新领域
  • 批准号:
    2235233
  • 财政年份:
    2023
  • 资助金额:
    $ 15.34万
  • 项目类别:
    Continuing Grant
Conference: Texas Analysis and Mathematical Physics Symposium 2024
会议:2024 年德克萨斯分析与数学物理研讨会
  • 批准号:
    2331234
  • 财政年份:
    2023
  • 资助金额:
    $ 15.34万
  • 项目类别:
    Standard Grant
Workshop on Trends in Soliton Dynamics and Singularity Formation for Nonlinear Dispersive PDEs
非线性色散偏微分方程孤子动力学和奇点形成趋势研讨会
  • 批准号:
    2230164
  • 财政年份:
    2022
  • 资助金额:
    $ 15.34万
  • 项目类别:
    Standard Grant

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Bounds and Asymptotic Dynamics for Nonlinear Evolution Equations
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  • 批准号:
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  • 财政年份:
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