The Whitham Equations and Their Solutions

数学科学：Whitham方程及其解摘要项目负责人将研究描述非线性色散振荡宏观结构的Whitham方程的各种问题。特别是，首席研究员将研究(1)一个空间维度的多相惠瑟姆方程，(2)几个空间维度的惠瑟姆方程。第一个项目的主要兴趣是单相振荡的相互作用以及双相和更高相振荡的产生和传播。第二个项目的基本目标是了解零相位Whitham解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的。该项目的研究成果将对跨学科工作产生广泛的影响。一个空间变量的多相Whitham方程在非线性色散振荡的零色散极限理论和调制理论中都起着重要的作用。它们在光纤脉冲传输中也有应用。若干空间维度上的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解具有内在联系。

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.