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

2015/01/02 by B. Fritzsche, M. A. Kaashoek, B. Kirstein +1 · 1 citation
Mathematics · Physics and Astronomy · #Dirac (video compression format) #Function (biology) #Integrable system #Inverse #Inverse problem #Isotropy #Matrix (chemical analysis) #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics #Rational function #Spectral Theory in Mathematical Physics #math.AP #math.CA #math.DS #math.OC #math.SP #msc:34B20 #msc:35Q55 #msc:37K10 #msc:39A06 #msc:93B28

paper · pdf · doi:10.1002/mana.201500069

published as Math. Nachr. 289 (2016) 1792-1819

arxiv created 2015/01/02 · openalex publication_date 2016/01/25 · openalex created_date 2016/06/24 · arxiv updated 2016/11/03 · openalex updated_date 2026/08/05

Abstract

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.

Citations

• ASSOCIATED STURM-LIOUVILLE SYSTEMS
• Quasideterminant solutions of the generalized Heisenberg magnet model
• Discrete skew self-adjoint canonical system and the isotropic Heisenberg magnet model

Cited by

• GBDT of discrete skew-selfadjoint Dirac systems and explicit solutions of the corresponding non-stationary problems

Related

• Skew-self-adjoint Dirac systems with a rectangular matrix potential: Weyl theory, direct and inverse problems
• Discrete Dirac system: rectangular Weyl functions, direct and inverse problems
• Recovery of the Dirac system from the rectangular Weyl matrix function
• Inverse problems for self-adjoint Dirac systems: explicit solutions and stability of the procedure
• Dirac systems with locally square-integrable potentials: direct and inverse problems for the spectral functions