Meromorphic Solutions of Differential Equations and Algebro-Geometric Differential Operators

F.Gesztesy和PI最近的一项发现揭示了KdV方程与相关的常线性微分方程组只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。预计这种关系与更一般的可积系统有关，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在这方面起着重要的作用，它假定了相关线性方程的位势的作用。PI计划对阿贝尔函数理论进行扩展，以包括奇异代数曲线上的函数，因为这是完全刻画代数几何势所必需的。与KdV方程相联系的代数几何势也给出了拟周期、非自伴微分算子的第一个容易处理的例子。可积系统，特别是KdV方程在数学物理中扮演着重要的角色，因为它们被用作各种现象的模型，例如在力学、流体力学和非线性光学中。即使对于不可积系统，把它们看作是可积系统的扰动通常也是有效的。同样，逆谱理论是应用数学的核心部分，因为它在从量子理论到计算机层析成像的各个领域都是基础。因此，这一研究路线在过去和未来都受到了极大的关注。这个项目是这一努力的一部分。

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.