Skew-selfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations

• Published in: Mathematische Nachrichten 2016-10, Vol.289 (14-15), p.1792-1819
• Main Authors: Fritzsche, B., Kaashoek, M. A., Kirstein, B., Sakhnovich, A. L.
• Format: Article
• Language: English
• Subjects: 34B20; 35Q55; 37K10; 39A06; 93B28; continuous Dirac system; Control theory; direct problem; discrete Dirac system; explicit solution; generalized discrete Heisenberg magnet model; inverse problem; Inverse problems; pseudo-exponential potential; rational matrix function; realization; rectangular matrix potential; Weyl function; Weyl theory
• ISSN: 0025-584X; 1522-2616
• DOI: 10.1002/mana.201500069

In this paper we study direct and inverse problems for discrete and continuous skew‐selfadjoint Dirac systems with rectangular (possibly non‐square) pseudo‐exponential potentials. For such a system the Weyl function is a strictly proper rational matrix function and any strictly proper rational matrix function appears in this way. In fact, extending earlier results, given a strictly proper rational matrix function we present an explicit procedure to recover the corresponding potential using techniques from mathematical system and control theory. We also introduce and study a nonlinear generalized discrete Heisenberg magnet model, extending earlier results for the isotropic case. A large part of the paper is devoted to the related discrete systems of which the pseudo‐exponential potential depends on an additional continuous time parameter. Our technique allows us to obtain explicit solutions for the generalized discrete Heisenberg magnet model and evolution of the Weyl functions.