Skew‐selfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations
Copyright policy )Skew‐selfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations
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.
- Leipzig University Germany
- Vrije Universiteit Amsterdam Netherlands
- University of Vienna Austria
Inverse problems for PDEs, rational matrix function, realization, 34B20, Dirac system, rectangular matrix potential, generalized discrete Heisenberg magnet model, 101002 Analysis, Dynamical Systems (math.DS), 37K10, CANONICAL SYSTEMS, Weyl theory and its generalizations for ordinary differential equations, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, direct problem, 34B20, 35Q55, 37K10, 39A06, 93B28, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Weyl theory, continuous Dirac system, Mathematics - Dynamical Systems, Mathematics - Optimization and Control, Spectral Theory (math.SP), explicit solution, OPERATORS, DISCRETE, Weyl function, NLS equations (nonlinear Schrödinger equations), Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.), 101001 Algebra, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Heisenberg magnet, 93B28, 39A06, discrete Dirac system, 35Q55, MATRIX FUNCTIONS, pseudo-exponential potential, Mathematics - Classical Analysis and ODEs, Optimization and Control (math.OC), FORMULAS, inverse problem, Discrete version of topics in analysis, NONLINEAR EQUATIONS, Analysis of PDEs (math.AP)
Inverse problems for PDEs, rational matrix function, realization, 34B20, Dirac system, rectangular matrix potential, generalized discrete Heisenberg magnet model, 101002 Analysis, Dynamical Systems (math.DS), 37K10, CANONICAL SYSTEMS, Weyl theory and its generalizations for ordinary differential equations, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, direct problem, 34B20, 35Q55, 37K10, 39A06, 93B28, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Weyl theory, continuous Dirac system, Mathematics - Dynamical Systems, Mathematics - Optimization and Control, Spectral Theory (math.SP), explicit solution, OPERATORS, DISCRETE, Weyl function, NLS equations (nonlinear Schrödinger equations), Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.), 101001 Algebra, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Heisenberg magnet, 93B28, 39A06, discrete Dirac system, 35Q55, MATRIX FUNCTIONS, pseudo-exponential potential, Mathematics - Classical Analysis and ODEs, Optimization and Control (math.OC), FORMULAS, inverse problem, Discrete version of topics in analysis, NONLINEAR EQUATIONS, Analysis of PDEs (math.AP)
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