Skew-Self-Adjoint Dirac System with a Rectangular Matrix Potential: Weyl Theory, Direct and Inverse Problems

Fritzsche, Bernd; Kirstein, Bernd; Roitberg, Inna Ya.; Sakhnovich, Alexander

doi:10.1007/s00020-012-1997-1

Cited by 19 publications

(29 citation statements)

References 54 publications

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“…In a way, which is similar to the m 1 = m 2 case (see [39]) and to the focusing NLS equation case (see [18,31]) we obtain our main result in this section. …”

Section: Remark 24supporting

confidence: 69%

Nonlinear Schrödinger equation in a semi-strip: Evolution of the Weyl–Titchmarsh function and recovery of the initial condition and rectangular matrix solutions from the boundary conditions

Sakhnovich

2015

Journal of Mathematical Analysis and Applications

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“…In a way, which is similar to the m 1 = m 2 case (see [39]) and to the focusing NLS equation case (see [18,31]) we obtain our main result in this section. …”

Section: Remark 24supporting

confidence: 69%

Nonlinear Schrödinger equation in a semi-strip: Evolution of the Weyl–Titchmarsh function and recovery of the initial condition and rectangular matrix solutions from the boundary conditions

Sakhnovich

2015

Journal of Mathematical Analysis and Applications

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“…Following the definition of a Weyl function for Dirac systems with square potentials (see also for the case of non‐square potentials) we say that a meromorphic function φ satisfying is a Weyl function of the system whenever it satisfies the inequality

\begin{matrix} true \\ left \int_{0}^{\infty} ([], I_{m_{1}}, φ, {(z)}^{*}) u {(x, z)}^{*} u (x, z) ([], \begin{matrix} I_{m_{1}} \\ φ (z) \end{matrix}) d x < \infty (z \in {double-struckC}_{M}) . \end{matrix}

Here u is the fundamental solution of normalized by

u (0, z) = I_{m}

. We note that for skew‐selfadjoint Dirac systems Weyl functions have been introduced in in an equivalent but different way. However, Proposition and Corollary 2.8 from immediately yield the following result concerning the direct problem.…”

Section: Continuous Case: Direct and Inverse Problemmentioning

confidence: 99%

“…We note that for skew‐selfadjoint Dirac systems Weyl functions have been introduced in in an equivalent but different way. However, Proposition and Corollary 2.8 from immediately yield the following result concerning the direct problem. Proposition Let hold.…”

Section: Continuous Case: Direct and Inverse Problemmentioning

confidence: 99%

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

Fritzsche

Kaashoek²,

Kirstein

et al. 2016

Mathematische Nachrichten

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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):

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“…Weyl theory of skew-selfadjoint Dirac systems was studied in the papers [22,28,46,48]. Some further references as well as the results of this subsection are contained in [58,Ch.3].…”

Section: Theorem 26 [56]mentioning

confidence: 99%

Evolution of Weyl Functions and Initial-Boundary Value Problems

Sakhnovich

2016

Math. Model. Nat. Phenom.

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This review is dedicated to some recent results on Weyl theory, inverse problems, evolution of the Weyl functions and applications to integrable wave equations in a semistrip and quarter-plane. For overdetermined initial-boundary value problems, we consider some approaches, which help to reduce the number of the initial-boundary conditions. The interconnections between dynamical and spectral Dirac systems, between response and Weyl functions are studied as well.

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Skew-Self-Adjoint Dirac System with a Rectangular Matrix Potential: Weyl Theory, Direct and Inverse Problems

Cited by 19 publications

References 54 publications

Nonlinear Schrödinger equation in a semi-strip: Evolution of the Weyl–Titchmarsh function and recovery of the initial condition and rectangular matrix solutions from the boundary conditions

Nonlinear Schrödinger equation in a semi-strip: Evolution of the Weyl–Titchmarsh function and recovery of the initial condition and rectangular matrix solutions from the boundary conditions

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

Evolution of Weyl Functions and Initial-Boundary Value Problems

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