Mathematical Sciences: Some Analytical Aspects of the Theory of the Integrable Systems

数学科学：可积系统理论的一些分析方面

• 批准号: 9315964
• 负责人: Alexander Its
• 金额: $ 4.22万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-08-01 至 1996-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9315964&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Some; Analytical; Aspects

Its 9300638 The research supported by this award falls into three related areas in the mathematical theory of nonlinear integrable equations, that is equations which admit a Lax-pair representation. The work derives from the first use of inverse scattering methods in solving nonlinear equations and has evolved into an important branch of mathematical physics. The analytic aspects of inverse scattering include a wide range of problems arising in the study of the solutions of integrable equations. For example, a large class of explicit quasiperiodical finite-gap solutions, as well as the comprehensive description of the long-time behavior of the solution of the Cauchy problem for an integrable system, can be obtained by inverse scattering. This project focuses on the application of the Riemann-Hilbert problem technique to the asymptotic analysis of the correlation functions of the quantum exactly solvable models. Continuing work will also be done on the development of the isomonodromy technique for the investigation of the Painleve-type ordinary differential equations and their application to the matrix model of 2D quantum gravity. A third line of investigation will be concerned with boundary value problems of integrable equations within the framework of the finite-gap integration and Riemann-Hilbert problem approach. Differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. The work represented in this project has close relationships with theoretical physics as well as the traditional synergisms with studies of wave phenomena ***

它的9300638该奖项支持的研究属于非线性集成方程数学理论的三个相关领域，即允许松弛对代表的方程式。该工作源自首次使用反散射方法来求解非线性方程，并演变为数学物理学的重要分支。反向散射的分析方面包括在可集成方程的解决方案的研究中引起的广泛问题。 例如，可以通过反向散射来获得大量的明确的准二维有限差距解决方案，以及对Cauchy问题解决方案的长期行为的全面描述。 该项目的重点是将Riemann-Hilbert问题技术应用于对量子恰好可解决模型的相关函数的渐近分析。 还将继续进行持续的工作，以开发同构型技术，以研究Painleve型普通微分方程及其在2D量子重力的矩阵模型中的应用。第三道调查将与有限间隙集成和Riemann-Hilbert问题方法的框架内的可集成方程的边界价值问题有关。 微分方程构成了物理世界数学建模的基础。 数学分析的作用并不是创建方程，而是提供有关解决方案的定性和定量信息。 这可能包括有关独特性，光滑和成长的问题的答案。 此外，分析通常会开发出解决这些近似值准确性的解决方案和估计值的方法。该项目所代表的工作与理论物理学以及传统的协同作用与波浪现象的研究***有着密切的关系。