Mathematical Sciences: Nonlinear Oscillations/Completely Integrable Systems

数学科学：非线性振荡/完全可积系统

• 批准号: 8702526
• 负责人: Stephanos Venakides
• 金额: $ 8万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-06-01 至 1990-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8702526&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Nonlinear; Oscillations; Completely

Professor Venakides intends to study mathematical problems of oscillation and wave propagation. He wants to determine the nature of oscillations in waveforms which are governed by a nonlinear partial differential equation of Schrodinger type. The interesting part of this work consists of studying the passage to the limit when the dispersive term becomes very small. Problems of this type were studied for the Korteweg-DeVries equation by Peter Lax, Henry McKean and Eugene Trubowitz of Courant Institute and by Herman Flaschka and others from the University of Arizona. Venakides brings new techniques to his study and wants to derive an integral equation which would characterize the solution in the limit case. He also wants to perform a similar study for a system of nonlinear ordinary differential equations known as Toda Lattice. Other projects described by him include an inverse problem for a Schrodinger equation, study of equations of granular flow and the work on the understanding of certain procedure for stochastic equations with a nonlinear operator. This research falls into the general area of studies in nonlinear partial differential equations which describe wave propagation phenomena such as solitary waves, acoustic waves, shocks and wavefronts, optical communications and laser technology. This particular research is intended to add to the knowledge base underlying such physical phenomena. The nonlinear partial differential equations of the type investigated in this project are important mathematical tools for engineering problems encountered in wave propogation.

Venakides教授打算研究振荡和波传播的数学问题。 他想确定波形中振荡的性质，该波形由施罗宾格类型的非线性部分微分方程控制。 这项工作的有趣部分包括在分散术语变得很小时研究段落到极限。 Peter Lax，Courant Institute的Peter Lax，Henry McKean和Eugene Trubowitz以及Herman Flaschka和其他亚利桑那大学的其他问题研究了这种类型的问题。 Venakides为他的研究带来了新的技术，并希望得出一个积分方程，该方程将在极限情况下进行特征。 他还想为一种称为Toda晶格的非线性普通微分方程系统进行类似的研究。 他描述的其他项目包括用于施罗宾格方程的反问题，颗粒流程方程的研究以及对使用非线性操作员对随机方程的某些程序的理解的工作。 这项研究属于非线性偏微分方程的一般研究领域，这些方程描述了波浪传播现象，例如孤立波，声波，冲击和波前，光学通信和激光技术。这项特殊的研究旨在增加这种物理现象的基础知识基础。 该项目研究类型的非线性偏微分方程是波动中遇到的工程问题的重要数学工具。