0405528/0405526 Gesztesy/Clark University of Missouri Columbia and Rolla
Collaborative Research: Systems of Ordinary Differential Equations - Inverse and Non-Self-Adjoint Problems
Research is proposed in two areas of systems of ordinary differential equations pertaining to inverse spectral problems and a class of non-self-adjoint singular Dirac-type boundary value problems. The research problems proposed lead to important applications in connection with completely integrable nonlinear evolution equations and to applications in soliton based optical communication systems. The first area is concerned with inverse spectral problems with emphasis on characterizing isospectral manifolds for self-adjoint matrix-valued Schroedinger and Dirac-type operators with periodic (and certain classes of quasi-periodic) coefficients. The techniques involved comprise matrix-valued Herglotz functions, inverse spectral theory, uniqueness theorems of Borg and Hochstadt-type, and pencils of matrices and their factorizations. The second area is concerned with spectral theory for a non-self-adjoint singular boundary value problem associated with a particular Dirac-type operator. The latter permits the existence of Weyl-Titchmarsh-type solutions for any point in its resolvent set under the most general hypothesis of merely local integrability of the potential coefficient. This property has not previously been observed in non-self-adjoint boundary value problems and hence makes this Dirac-type operator a model operator of particular interest. The common thread through all the problems proposed is the use of (matrix-valued) Weyl-Titchmarsh-type functions which encode all spectral information of the underlying Schroedinger and Dirac-type systems.
Inverse spectral theory for self-adjoint Schroedinger and Dirac-type boundary value problems, as proposed in the first part of this proposal, is one of the pillars of applications of spectral theory to the applied sciences including theoretical physics (quantum physics), geophysics (seismology), medicine (tomography), etc. As such, it is an integral part of modern applied mathematics. In addition, this first part permits applications to completely integrable systems, especially to soliton equations such as the nonabelian Korteweg-de Vries and the defocusing nonlinear Schroedinger hierarchies of evolution equations. Completely integrable systems of this type, a rapidly developing field in pure and applied mathematics especially since the second half of the 20th century, have widespread and multifaceted applications which include shallow water wave modelling, various aspects of nonlinear optics, and problems in condensed matter physics. On the other hand, many concrete applications of completely integrable systems naturally lead to non-self-adjoint boundary value problems. A prime example would be the area of nonlinear optics as modelled by the focusing nonlinear Schroedinger equation. The latter is 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 spectral theory) for such non-self-adjoint Dirac operators is still in its infancy, we propose a new model for a soliton based optical communication system based upon our proposed study of a special class of soliton potentials relative to a periodic background potential.
