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Vector functional-difference equation in electromagnetic scattering. (English) Zbl 1059.78013

The paper deals with a vector functional-difference equation: given a matrix-valued function \(G(\sigma)\) and a vector-valued function \(g(\sigma)\), \(\operatorname{Re}\sigma=\omega\), find a vector-valued function \(\Phi(s)\) which is analytic in the strip \(\omega-h<\operatorname{Re} s <\omega\) and satisfy the relation \[ \Phi(\sigma)=G(\sigma)\Phi(\sigma-h)+g(\sigma),\quad \operatorname{Re} \sigma =\omega. \] The authors propose a procedure for exact solution of this problem in the case where \(G(\sigma)\) has a certain special structure. The method is based on converting the problem first to a vector Riemann-Hilbert problem on a system of open curves and subsequently to a Riemann-Hilbert problem on a contour on a hyper-elliptic surface formed from two copies of the cut complex plane. Solving the associated Jacobi inversion problem, the authors construct the general solution of the Riemann-Hilbert problem on the surface and, afterwards, the general solution of the initial functional-difference equation.
The proposed technique is applied to the problem of electromagnetic scattering by an anisotropic impedance half- plane. A closed-form solution is presented, which correspond to a hyper-elliptic surface of genus three.

MSC:

78A45 Diffraction, scattering
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30E25 Boundary value problems in the complex plane
30F30 Differentials on Riemann surfaces
34K10 Boundary value problems for functional-differential equations
39B72 Systems of functional equations and inequalities
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