Integrable systems, the Whitham equations and conformal maps

可积系统、Whitham 方程和共形映射

• 批准号: 0104621
• 负责人: Igor Krichever
• 金额: $ 14.51万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104621&HistoricalAwards=false
• 关键词: Integrable; systems; Whitham; equations; conformal

Abstract for DMS - 0104621The algebraic-geometric theory of soliton equations developed in the middle seventies has become one of the most powerful tools in the theory of integrablemodels, and has had enormous influence on many branches of mathematics and theoretical physics. The main objective of the present projectis a further development of this approach to integration of non-linear equations, models of solid state physics, and models of quantum field theories.An immediate goal is to construct integrable models correspondingto various supersymmetric gauge Seiber-Witten models, which can be instrumental in investigating key physical issues such as duality, the renormalization group, or instanton corrections.Another goal of the project is a classification of commuting difference operators. Particular attention will be paid to connection of this theory with the Hitchin system which is central in modern theory of the moduli space of vector bundles over algebraic curve.Recent progress in understanding non-perturbative structures in supersymmetric gauge theories has shed new insight upon the role of integrable structures in modern theoretical physics.The nineteenth century saw many aspects of geometry and analysis,particularly the development of Abelian functions, drawn together in the study of integrable systems. The works of Jacobi, Abel, Riemann and Weierstrass enabled a number of important integrableproblems of mechanics and physics to be solved. The modern discovery of soliton theory has led to a renewed interest to the theory of integrable systems. Nonlinear phenomena could now be treated, and the ever-growing interest in this theory is connected with the fact that it is applicable to equations which possess a remarkable universality property. They arise in thedescription of the most diverse phenomena in plasma physics, the theory ofelementary particles, the theory of superconductivity and in nonlinear optics.Geometry and algebraic geometry, functional equations and special functions,Lie algebras and groups all come together in their study. This ubiquity of integrable systems together with the beautiful structures that underly them has been confirmed recently by discovery of unexpected relations between the Whitham perturbation theory of soliton equations developed in earlier works of the author and the Riemann mapping theorem. It seems urgent to develop methods which can identify various Whitham hierarchies with conformal maps for more general types of domains. A range of possible applicationsinclude flows in porous media, fundamental theory of pattern formation.

孤子方程的代数-几何理论是七十年代中期发展起来的，它已成为可积模型理论中最有力的工具之一，并对数学和理论物理的许多分支产生了巨大的影响。本项目的主要目标是进一步发展这种方法，将其应用于非线性方程、固体物理模型和量子场论模型的积分，近期目标是构造对应于各种超对称规范Seiber-Witten模型的可积模型，这将有助于研究诸如对偶性、重整化群、该项目的另一个目标是对易差分算子进行分类。特别注意这个理论与Hitchin系统的联系，Hitchin系统是现代代数曲线上向量丛模空间理论的中心。最近在理解超对称规范理论中的非微扰结构方面的进展，对可积结构在现代理论物理中的作用有了新的认识。19世纪见证了几何和分析的许多方面，特别是阿贝尔函数的发展，在可积系统的研究中一起绘制。Jacobi、Abel、Riemann和Weierstrass的工作使一些重要的力学和物理学的可积问题得以解决。近代孤子理论的发现使人们对可积系统理论产生了新的兴趣。现在可以处理非线性现象，对这一理论日益增长的兴趣与它适用于具有显着普适性质的方程这一事实有关。它们出现在等离子体物理学、基本粒子理论、超导理论和非线性光学中最多样化现象的描述中。几何和代数几何、函数方程和特殊函数、李代数和群都在其研究中聚集在一起。这种无处不在的可积系统连同美丽的结构，他们已经证实最近发现的意想不到的关系之间的Whitham微扰理论的孤子方程在早期的作品作者和黎曼映射定理。对于更一般类型的域，用保角映射来识别各种Whitham族的方法似乎是迫切需要的。其应用范围包括多孔介质中的流动、井网形成的基础理论等.