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

摘要：0405528/0405526 Gesztesy/Clark密苏里州哥伦比亚大学和Rolla合作研究：常微分方程系统-逆和非自伴问题研究提出了两个领域的常微分方程系统有关的逆谱问题和一类非自伴奇异Dirac型边值问题。所提出的研究问题在完全可积的非线性演化方程和基于光孤子的光通信系统中有着重要的应用。第一个领域是关于逆谱问题，重点是刻画具有周期（和某些类拟周期）系数的自伴矩阵值Schroedinger和Dirac型算子的等谱流形，涉及的技术包括矩阵值Herglotz函数，逆谱理论，博格和Hochstadt型唯一性定理，矩阵束及其分解。第二部分是研究一类特殊Dirac型算子的非自伴奇异边值问题的谱理论。后者允许存在Weyl-Titchmarsh型解决方案的任何点在其预解集的最一般的假设下，只是局部可积的潜在系数。 这一性质以前没有被观察到在非自伴边值问题，因此使这Dirac型算子模型运营商特别感兴趣。贯穿所有提出的问题的共同思路是使用（矩阵值）Weyl-Titchmarsh型函数，它编码了Schroedinger和Dirac型系统的所有谱信息.本文第一部分提出的自伴Schroedinger和Dirac型边值问题的逆谱理论，是谱理论在包括理论物理在内的应用科学中的重要应用之一（量子物理学）、物理学（地震学）、医学（断层摄影术）等。因此，它是现代应用数学的一个组成部分。此外，第一部分还允许应用于完全可积系统，特别是孤子方程，如非交换的Korteweg-deVries和散焦的非线性Schroedinger族的演化方程。这类完全可积系统在理论数学和应用数学领域发展迅速，特别是世纪后半期以来，在浅水波模拟、非线性光学的各个方面以及凝聚态物理问题中有着广泛而多方面的应用。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。一个最好的例子是非线性光学领域，它是由聚焦非线性薛定谔方程模拟的。后者与非自伴Dirac型算子密切相关，这是本建议第二部分的主要研究对象。（尤其是逆谱理论）对于这种非自伴Dirac算子的研究还处于起步阶段，基于我们对一类特殊的孤子势的研究，提出了一种新的基于孤子的光通信系统模型潜力

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.