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Mathematical Sciences: Study of Nonlinear Problems by the IST Method

数学科学：用 IST 方法研究非线性问题

基本信息

批准号：
9111611
负责人：
Athanassios Fokas
金额：
$ 7万
依托单位：
Clarkson University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
1992
资助国家：
美国
起止时间：
1992-04-01 至 1996-03-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9111611&HistoricalAwards=false
关键词：
Mathematical Sciences Study Nonlinear Problems

项目摘要

This project supports the collaboration between a group of U.S. scientists at Clarkson University and a group of Russian scientists from University of Leningrad. The research concerns the study of nonlinear partial differential equations by the Inverse Scattering Transform (IST) Method. The IST method has been a successful and powerful method of solution in the 1+1 (one space and one time dimension) setting. The power of the IST method comes from the fact that the solution of certain nonlinear partial differential equations can be reduced to the solution of the inverse problem for linear ordinary differential equations, and such an inverse problem can usually be formulated as a Riemann- Hilbert problem. As a result, the solution of certain nonlinear partial differential equations have been obtained simply by using the methods to solve linear differential equations. The generalization of the IST method to N+1, where N is greater than 1, has not been as successful and so far some results have only been obtained for N=2. The Russian team at Leningrad is noted as one of the top teams in the world in this area of science. A collaboration would have significant scientific advantage to U.S. scientists. The science supported in this US-USSR collaboration is in the general area of partial differential equations. This research topic concerns the equations of motion, fluid flow, physics, etc. that model our physical universe.

该项目支持克拉克森大学的一组美国科学家和列宁格勒大学的一组俄罗斯科学家之间的合作。研究了用逆散射变换(IST)方法研究非线性偏微分方程组。在1+1(一个空间和一个时间维度)的背景下，IST方法已经成为一种成功而有力的解决方法。IST方法的强大之处在于，某些非线性偏微分方程组的解可以归结为线性常微分方程逆问题的解，而这样的逆问题通常可以表示为Riemann-Hilbert问题。由此，利用线性微分方程解的方法，简单地得到了一类非线性偏微分方程组的解。将IST方法推广到N+1(其中N大于1)并不成功，到目前为止只在N=2的情况下得到了一些结果。列宁格勒的俄罗斯团队被认为是世界上这一科学领域的顶级团队之一。合作将对美国科学家产生重大的科学优势。这次美苏合作所支持的科学是偏微分方程式的一般领域。这个研究主题涉及对我们的物理宇宙建模的运动方程、流体流动、物理学等。