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

Abstract

In this paper we study direct and inverse problems for discrete and continuous time skew-selfadjoint Dirac systems with rectangular (possibly non-square) pseudo-exponential potentials. For such a system the Weyl function is a strictly proper rational rectangular 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 time systems of which the pseudo-exponential potential depends on an additional continuous time parameter. Our techniques allows us to obtain explicit solutions for the generalized discrete Heisenberg magnet model and evolution of the Weyl functions.

References in corpus (2)

• Bidifferential Calculus Approach to AKNS Hierarchies and Their Solutions
• Discrete skew selfadjoint canonical systems and the isotropic Heisenberg magnet model

Cited by in corpus (4)

• Skew-selfadjoint Dirac systems: stability of the procedure of explicit solving the inverse problem
• Scattering for general-type Dirac systems on the semi-axis: reflection coefficients and Weyl functions
• Discrete Dirac systems on the semiaxis: rational reflection coefficients and Weyl functions
• Continuous and discrete dynamical Schrödinger systems: explicit solutions