Mathematical Sciences: Scattering and Stability of NonlinearWaves

数学科学：非线性波的散射和稳定性

• 批准号: 9201717
• 负责人: Michael Weinstein
• 金额: $ 6.5万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-06-01 至 1995-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9201717&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Scattering; Stability; NonlinearWaves

Work on this mathematics project is grouped into three main areas: (a) nonlinear scattering and orbital asymptotic stability in infinite dimensional Hamiltonian systems, (b) transitions to instability and (c) vector Zakharov and nonlinear Schrodinger equations. The first class of problems is concerned with the long-time behavior of solutions of nonlinear dispersive systems in terms of a nonlinear bound state channel (modeling nondecaying behavior) and a dispersive radiation field. The governing equations have the structure of a finite dimensional dynamical system which is coupled to the infinite dimensional dispersive radiation field. In the second, a new theory for a class of eigenvalue problems will be applied to nonlinear equations modeling long wave propagation in dispersive media. An advanced formulation is being developed which is independent of the dynamical systems framework. Applications to nonlocal equations arising in the study of fluids and plasmas will be sought. The third goal is to study the existence and properties of nonlinear bound states of the vector Zakharov and Schrodinger equations. These systems are a better approximation of the physics of waves in a collisionless plasma. Associated with these equations are believed to be phenomena not present in scalar systems, such as non-isotropic ground states, which may participate in the dynamics of singularity formation. Differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.

这个数学项目的工作分为三个主要部分 （a）非线性散射和轨道渐近稳定性 在无限维哈密顿系统中，（B）过渡到 不稳定性与（c）矢量Zakharov和非线性薛定谔 方程 第一类问题是关于 非线性色散方程组解的长时间行为 根据非线性束缚态信道（建模非衰减 行为）和色散辐射场。 理事 方程具有有限维动力学结构 系统耦合到无限维色散 辐射场在第二章中，我们提出了一个新的理论， 特征值问题将应用于非线性方程 模拟色散介质中的长波传播。 一种先进 正在开发的配方是独立的， 动力系统框架 非局部方程的应用 在流体和等离子体的研究中产生的。的 第三个目标是研究非线性系统的存在性和性质 矢量Zakharov和薛定谔方程的束缚态。 这些系统是波物理学的更好近似 在无碰撞的等离子体中 与这些方程相关联的是 被认为是标量系统中不存在的现象，例如 非各向同性基态，其可以参与 奇点形成的动力学 微分方程是数学的主干 物理科学中的建模。 现象涉及 连续变化，如在运动中看到的，材料和 已知能量服从某些一般定律， 可以用相互作用和关系来表达 偏导数之间的关系 数学的关键作用不是 来陈述这些关系，而是为了提取定性的 和量化的意义。