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

其9300638 该奖项支持的研究分为三个相关领域的非线性可积数学理论的福尔斯 方程，即允许Lax对表示的方程。这项工作源于第一次使用逆散射方法求解非线性方程组， 成为数学物理学的一个重要分支。逆散射的分析方面包括广泛的问题 在研究可积方程的解时产生的。 例如，一大类明确的准周期有限间隙的解决方案，以及全面的描述的柯西问题的解决方案的长期行为的可积系统，可以通过逆散射。 本计画主要应用Riemann-Hilbert问题技巧于量子精确可解模型关联函数的渐近分析。 还将继续开展工作，发展用于研究Painleve型常微分方程及其在二维量子引力矩阵模型中的应用的等势线技术。第三条线的调查将关注的框架内的有限间隙积分和黎曼-希尔伯特问题的方法可积方程的边值问题。 微分方程是物理世界数学建模的基础。 数学分析的作用与其说是建立方程，不如说是提供关于解的定性和定量信息。 这可能包括关于独特性，平滑性和增长的问题的答案。 此外，分析经常发展出解的近似方法和对这些近似的准确性的估计。该项目所代表的工作与理论物理以及与波动现象研究的传统协同作用有着密切的关系 *

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 ***