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

摘要/ Abstract摘要：Gesztesy/Clark University of Missouri and rolla合作研究：常微分方程系统-反与非自伴随问题研究提出了两个领域的常微分方程系统的反谱问题和一类非自伴随奇异dirac型边值问题。所提出的研究问题在与完全可积非线性演化方程相关的研究和基于光通信系统的应用中具有重要的应用价值。第一个领域是关于逆谱问题，重点是描述具有周期（和某些类准周期）系数的自伴随矩阵值薛定谔算子和狄拉克算子的等谱流形。所涉及的技术包括矩阵值赫格罗兹函数、逆谱理论、Borg和hochstadt型唯一性定理、矩阵及其分解。第二部分讨论了与特定ardirac型算子相关的非自共轭边值问题的谱理论。后者在势系数仅局部可积的最一般假设下，允许其解集中任意点的weyl - titchmarsh型解的存在。这个性质以前没有在非自伴随边值问题中观察到，因此使这个狄拉克型算子成为一个特别感兴趣的模型算子。贯穿所有问题的共同线索是使用（矩阵值）weyl - titchmarsh型函数，该函数编码底层薛定谔和狄拉克类型系统的所有光谱信息。本文第一部分提出的自伴随薛定谔和狄拉克型边值问题的逆谱理论是将谱理论应用于理论物理（量子物理）、地球物理（地震学）、医学（断层扫描）等应用科学的支柱之一。因此，它是现代应用数学的一个组成部分。此外，第一部分允许应用于完全可积分系统，特别是孤子方程，如非阿贝尔Korteweg-deVries和离焦非线性薛定谔进化方程层次。这种类型的完全可积系统，特别是自20世纪下半叶以来，在纯数学和应用数学中迅速发展，具有广泛和多方面的应用，包括浅水波建模、非线性光学的各个方面以及凝聚态物理问题。另一方面，完全可积系统的许多具体应用自然导致非自伴随边值问题。一个主要的例子是非线性光学领域，它由聚焦非线性薛定谔方程建模。后者与非自伴狄拉克型算子密切相关，狄拉克算子是本文第二部分研究的主要对象。虽然这种非自伴随狄拉克算子的一般谱理论（尤其是逆谱理论）仍处于起步阶段，但我们提出了一个基于孤立子的光通信系统的新模型，该模型基于我们对相对于周期背景势的一类特殊孤子势的研究。

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.