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

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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.