Collaborative Research: Systems of Ordinary Differential Equations - Inverse and Non-Self-Adjoint Problems

摘要：0405528/0405526密苏里州哥伦比亚大学和罗拉大学的Gesztesy/Clark合作研究：常微分方程组-逆和非自伴问题在两个领域的常微分方程组中提出了关于逆谱问题和一类非自伴奇异Dirac型边值问题的研究。所提出的研究问题在完全可积的非线性发展方程和基于孤子的光通信系统中具有重要的应用。第一个领域涉及逆谱问题，重点刻画了具有周期(和某些类拟周期)系数的自伴矩阵值薛定谔算子和Dirac算子的等谱流形.所涉及的技巧包括矩阵值Hergltz函数，逆谱理论，Borg和Hochstadt型唯一性定理，以及矩阵的铅笔及其分解.第二个区域涉及与特殊Dirac算子相关的非自伴奇异边值问题的谱理论。后者在势系数仅局部可积的最一般假设下，允许其预解集中的任意点存在Weyl-Titchmarsh型解。这一性质以前没有在非自伴边值问题中观察到，因此使这个Dirac-型算子成为一个特别感兴趣的模型算子。所有提出的问题的共同主线是使用(矩阵值)Weyl-Titchmarsh型函数，它编码了底层薛定谔和狄拉克类型系统的所有光谱信息。本提案第一部分提出的自伴薛定谔和狄拉克类型边值问题的逆谱理论是谱理论应用于应用科学的支柱之一，包括理论物理(量子物理)、地球物理(地震学)、医学(层析)等。因此，它是现代应用数学的一个组成部分。此外，这第一部分允许应用于完全可积系统，特别是孤子方程，如非阿贝尔Korteweg-DeVries和散焦的非线性薛定谔发展方程。这种类型的完全可积系统，特别是20世纪下半叶以来，在理论数学和应用数学领域迅速发展，有着广泛和多方面的应用，包括浅水波模型、非线性光学的各个方面以及凝聚态物理中的问题。另一方面，完全可积系统的许多具体应用自然会导致非自伴边值问题。一个最好的例子是由聚焦非线性薛定谔方程模拟的非线性光学领域。虽然非自伴Dirac算子的一般谱理论(特别是逆谱理论)仍处于起步阶段，但基于我们提出的一类相对于周期背景势的特殊孤子势的研究，我们提出了一种新的基于孤子的光通信系统模型。

Abstract: 0405528/0405526 Gesztesy/Clark University of Missouri Columbia and RollaCollaborative Research: Systems of Ordinary Differential Equations - Inverse and Non-Self-Adjoint Problems Research is proposed in two areas of systems of ordinarydifferential equations pertaining to inverse spectral problems and aclass of non-self-adjoint singular Dirac-type boundary valueproblems. The research problems proposed lead to importantapplications in connection with completelyintegrable nonlinear evolution equations and to applications insoliton based optical communication systems. The first area isconcerned with inverse spectral problems with emphasis oncharacterizing isospectral manifolds forself-adjoint matrix-valued Schroedinger and Dirac-type operators withperiodic (and certain classes of quasi-periodic) coefficients.The techniques involved comprise matrix-valued Herglotz functions,inverse spectral theory, uniqueness theorems of Borg andHochstadt-type, and pencils of matrices and their factorizations. Thesecond area is concerned with spectral theory for a non-self-adjointsingular boundary value problem associated with a particularDirac-type operator. The latter permits the existence ofWeyl-Titchmarsh-type solutions for any point in its resolvent setunder the most general hypothesis of merely local integrability ofthe potential coefficient. This property has not previously beenobserved in non-self-adjoint boundary value problems and hence makesthis Dirac-type operator a model operator of particular interest. Thecommon thread through all the problems proposed is the use of(matrix-valued) Weyl-Titchmarsh-type functions which encode allspectral information of the underlying Schroedinger and Dirac-typesystems.Inverse spectral theory for self-adjoint Schroedinger and Dirac-typeboundary value problems, as proposed in the first part of thisproposal, is one of the pillars of applications of spectral theory tothe applied sciences including theoretical physics (quantum physics),geophysics (seismology), medicine (tomography), etc. As such, it isan integral part of modern applied mathematics. In addition, thisfirst part permits applications to completely integrable systems,especially to soliton equations such as the nonabelian Korteweg-deVries and the defocusing nonlinear Schroedinger hierarchies ofevolution equations. Completely integrable systems of this type, arapidly developing field in pure and applied mathematics especiallysince the second half of the 20th century, have widespread andmultifaceted applications which include shallow water wave modelling,various aspects of nonlinear optics, and problems in condensed matterphysics. On the other hand, many concrete applications of completelyintegrable systems naturally lead to non-self-adjoint boundary valueproblems. A prime example would be the area of nonlinear optics asmodelled by the focusing nonlinear Schroedinger equation. The latteris intimately connected with a non-self-adjoint Dirac-type operator,the principal object of study in the second part of this proposal.While general spectral theory (and especially inverse spectraltheory) for such non-self-adjoint Dirac operators is still in itsinfancy, we propose a new model for a soliton based optical communicationsystem based upon our proposed study of a special class of solitonpotentials relative to a periodic background potential.