Mathematical Sciences: Nonlinear Oscillations/Completely Integrable Systems

维纳基德斯教授打算研究振荡和波传播的数学问题。他想确定由薛定谔类型的非线性偏微分方程所支配的波形中振荡的性质。这项工作的有趣部分包括研究当色散项变得很小时到达极限的通道。Courant研究所的Peter Lax、Henry McKean和Eugene Trubowitz以及亚利桑那大学的Herman Flaschka和其他人研究了Korteweg-DeVries方程的这类问题。Venakdes为他的研究带来了新的技术，他想要推导出一个积分方程来刻画极限情况下的解。他还想对一个被称为Toda格子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，以及用非线性算子理解随机方程的某些程序的工作。这项研究属于描述孤立波、声波、激波和波前、光通信和激光技术等波传播现象的非线性偏微分方程组的一般研究领域。这项特别的研究旨在增加这种物理现象背后的知识库。本课题所研究的这类非线性偏微分方程组是波传播中遇到的工程问题的重要数学工具。

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.