Mathematical Sciences: Nonlinear Oscillations/Completely Integrable Systems

Venakides教授打算研究数学问题， 振荡和波的传播。 他想确定 波形中的振荡由非线性部分 薛定谔型微分方程 有趣的是 工作包括研究通道的极限时，分散 这个词变得很小。 这类问题的研究是为了 Korteweg-DeVries方程Peter Lax，亨利麦基恩和尤金 Trubowitz和赫尔曼Flaschka等人， 亚利桑那大学 Venakides为他的研究带来了新技术 他想推导出一个积分方程， 极限情况下的解决方案。 他还想进行一项类似的研究 对于称为户田的非线性常微分方程系统， 格子。 他描述的其他项目包括一个逆问题， 一个薛定谔方程，研究的颗粒流方程和工作 关于随机方程的某些过程的理解 非线性算子 本文的研究属于非线性动力学研究的一般范畴，属于福尔斯的研究 描述波传播现象的偏微分方程 例如孤立波、声波、冲击和波前、光学 通信和激光技术。这项研究是 旨在增加这种物理基础的知识基础， 现象。 一类非线性偏微分方程 在这个项目中研究的是重要的数学工具， 波浪传播中遇到的工程问题。

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.