The Whitham Equations and Their Solutions

美国国家科学基金会奖摘要- DMS-0103849数学科学：惠瑟姆方程及其解摘要DMS-0103849田主要研究者将考虑各种有关惠瑟姆方程的问题，它描述了非线性色散振荡的宏观结构。特别是，首席研究员将研究（1）一维空间中的多相Whitham方程，以及（2）多个空间维度中的Whitham方程。第一个项目的主要兴趣是在单相振荡和双相和更高相位振荡的产生和传播的相互作用。第二个项目的基本目标是了解零相位Whitham解决方案如何在多维空间中发展奇点。所提出的方法将是分析和计算。该项目的成果将对跨学科工作产生广泛影响。单空间变量的多相Whitham方程在非线性色散振荡的零色散极限和调制理论中起着重要的作用。它们也可用于光纤中脉冲的传输。多维空间的Whitham方程与拓扑场论中的Landau-Ginzburg模型和超对称Yang-Mills理论中的Seiberg-Witten解有着内在的联系。

NSF Award Abstract - DMS-0103849Mathematical Sciences: The Whitham Equations and Their SolutionsAbstractDMS-0103849TianThe Principal Investigator will consider a variety of problems concerning the Whitham equations, which describe the macrostructure of nonlinear dispersive oscillations. In particular, the Principal Investigator will study (1) multiphase Whitham equations in one spatial dimension, and (2) Whitham equations in several spatial dimensions. The primary interest of the first project is in the interaction of single-phase oscillations and generation and propagation of double and higher phase oscillations. The basic goal of the second project is to understand how the zero phase Whitham solution develops singularities in several dimensional space. The proposed methods will be both analytical and computational. Results of this project will have broad impact in interdisciplinary work. The multiphase Whitham equations in one spatial variable play an essential role in both zero dispersion limit and modulation theories of nonlinear dispersive oscillations. They also have applications in the transmission of pulses in optical fibers. The Whitham equations in several spatial dimensions are intrinsically connected to Landau-Ginzburg models in topological field theory and the Seiberg-Witten solution in supersymmetric Yang-Mills Theory.