Meromorphic Solutions of Differential Equations and Algebro-Geometric Differential Operators

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A recent discovery by F. Gesztesy and the PI reveals a relationshipbetween the KdV equation and the condition that the associated ordinarylinear differential equations have only meromorphic solutions. The samerelationship exists also for the AKNS system. Both, the KdV equation andthe AKNS system, are completely integrable Hamiltonian systems. It isanticipated that the relationship pertains to more general integrablesystems and the major object of the project is to establish this. Animportant role in this respect is played by so called algebro-geometricsolutions of the integrable system under consideration which assume therole of potentials of the associated linear equations. The PI plans anexpansion of abelian function theory to include functions on singularalgebraic curves since this is required for a complete characterizationof algebro-geometric potentials. Algebro-geometric potentials associatedwith the KdV equation give also the first tractable examples ofquasi-periodic, non-selfadjoint differential operators. It is planned toinvestigate such operators and their spectrum and to develop an inversespectral theory for them.Integrable systems and, in particular, the KdV equation play animportant role in mathematical physics were they are used as models forvarious phenomena, e.g., in mechanics, hydrodynamics, and nonlinearoptics. Even for systems which are not integrable it is often fruitfulto consider them as perturbations of integrable systems. Likewise,inverse spectral theory is a central part of applied mathematics sinceit is fundamental in areas ranging from quantum theory to computertomography. Therefore this line of research has enjoyed an enormousamount of attention in the past and will so in the future. This projectis part of that endeavor.