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. MSC(2010):

Discrete Dirac systems on the semiaxis: rational reflection coefficients and Weyl functions

“…If S 0 > 0, the identity (2.20) holds and the pair {A, ϑ 1 } is controllable, then according to [6,Lemma 3.2] and [6, Proposition 3.6] we have det A = 0, S k > 0 and the matrices C k admit representation C k = U * k jU k from (1.5). That is, the sequence {C k } is well-defined and the corresponding system is a skew-self-adjoint Dirac system.…”

“…The preliminary definitions and results on the skew-self-adjoint discrete Dirac systems (SkDDS) (1.5) we take from [6] and sometimes from [8].…”

“…Discrete self-adjoint and skew-self-adjoint Dirac systems play an essential role in the study of Toeplitz matrices (and corresponding measures), of discrete integrable nonlinear equations (including isotropic Heisenberg magnet model) and of spectral theory of difference equations (see, e.g., [5,6,14,20,22] and references therein). Weyl-Titchmarsh theory of discrete systems is actively studied (see, e.g., [3,13,27,28] and various references therein).…”

“…Weyl-Titchmarsh theory of discrete systems is actively studied (see, e.g., [3,13,27,28] and various references therein). In particular, Weyl-Titchmarsh theory of discrete self-adjoint and skew-self-adjoint Dirac systems was studied in [6][7][8]14,18,24] (see also references therein). It is known that Weyl-Titchmarsh (or simply Weyl) functions of continuous Dirac systems on the semi-axis are closely related to the scattering data.…”

“…where the matrices U k are unitary and j is defined in (1.2). Direct and inverse problems (in terms of Weyl functions) were solved for systems (1.1), (1.2) in [7,18] and for systems (1.5) in [6,8]. In particular, in the case of rational Weyl matrix functions, direct and inverse problems were solved explicitly using our GBDT version [19,21,24] of the Bäcklund-Darboux transformation.…”