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

TL;DR

This paper develops explicit methods for solving direct and inverse problems for skew-selfadjoint Dirac systems with rational rectangular Weyl functions, and applies these to integrable wave equations and a generalized discrete Heisenberg magnet model.

Contribution

It provides an explicit procedure to recover potentials from rational Weyl functions and extends results to discrete and continuous time systems with applications to integrable models.

Findings

Explicit solutions for inverse problems in Dirac systems.

Recovery of potentials from rational Weyl functions.

Solutions for the generalized discrete Heisenberg magnet model.

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.