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。 他想要确定由薛定谔类型的非线性偏微分方程控制的波形振荡的性质。 这项工作有趣的部分在于研究当色散项变得非常小时如何达到极限。 Courant 研究所的 Peter Lax、Henry McKean 和 Eugene Trubowitz 以及亚利桑那大学的 Herman Flaschka 和其他人针对 Korteweg-DeVries 方程研究了此类问题。 Venakides 为他的研究带来了新技术，并希望导出一个积分方程来表征极限情况下的解。 他还想对称为 Toda Lattice 的非线性常微分方程组进行类似的研究。 他描述的其他项目包括薛定谔方程的反问题、颗粒流方程的研究以及理解带有非线性算子的随机方程的某些过程的工作。 这项研究属于非线性偏微分方程的一般研究领域，该方程描述了波传播现象，如孤立波、声波、冲击和波前、光通信和激光技术。这项特殊的研究旨在增加此类物理现象背后的知识库。 本项目研究的非线性偏微分方程类型是解决波浪传播中遇到的工程问题的重要数学工具。