Meromorphic Solutions of Differential Equations and Algebro-Geometric Differential Operators

微分方程和代数几何微分算子的亚纯解

• 批准号: 9970299
• 负责人: Rudi Weikard
• 金额: $ 3.83万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970299&HistoricalAwards=false
• 关键词: Meromorphic; Solutions; Differential; Equations; Algebro

A recent discovery by F. Gesztesy and the PI reveals a relationshipbetween the KdV equation and the condition that the associated ordinarylinear differential equations have only meromorphic solutions. The samerelationship exists also for the AKNS system. Both, the KdV equation andthe AKNS system, are completely integrable Hamiltonian systems. It isanticipated that the relationship pertains to more general integrablesystems and the major object of the project is to establish this. Animportant role in this respect is played by so called algebro-geometricsolutions of the integrable system under consideration which assume therole of potentials of the associated linear equations. The PI plans anexpansion of abelian function theory to include functions on singularalgebraic curves since this is required for a complete characterizationof algebro-geometric potentials. Algebro-geometric potentials associatedwith the KdV equation give also the first tractable examples ofquasi-periodic, non-selfadjoint differential operators. It is planned toinvestigate such operators and their spectrum and to develop an inversespectral theory for them.Integrable systems and, in particular, the KdV equation play animportant role in mathematical physics were they are used as models forvarious phenomena, e.g., in mechanics, hydrodynamics, and nonlinearoptics. Even for systems which are not integrable it is often fruitfulto consider them as perturbations of integrable systems. Likewise,inverse spectral theory is a central part of applied mathematics sinceit is fundamental in areas ranging from quantum theory to computertomography. Therefore this line of research has enjoyed an enormousamount of attention in the past and will so in the future. This projectis part of that endeavor.

F. Gesztesy 和 PI 最近的一项发现揭示了 KdV 方程与相关常线性微分方程仅具有亚纯解的条件之间的关系。 AKNS 系统也存在同样的关系。 KdV 方程和 AKNS 系统都是完全可积的哈密顿系统。预计这种关系适用于更一般的可积系统，该项目的主要目标是建立这种关系。在这方面，所考虑的可积系统的所谓代数几何解发挥着重要作用，该解假定相关线性方程的势的作用。 PI 计划扩展阿贝尔函数理论以包括奇异代数曲线上的函数，因为这是代数几何势的完整表征所必需的。与 KdV 方程相关的代数几何势也给出了准周期、非自伴微分算子的第一个易于处理的例子。计划研究这些算子及其谱，并为它们开发逆谱理论。可积系统，特别是 KdV 方程在数学物理中发挥着重要作用，因为它们被用作各种现象的模型，例如在力学、流体动力学、和非线性光学。即使对于不可积的系统，将它们视为可积系统的扰动通常也是富有成效的。同样，逆谱理论是应用数学的核心部分，因为它是从量子理论到计算机断层扫描等领域的基础。因此，这一领域的研究在过去和未来都受到了极大的关注。该项目是该努力的一部分。