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A genus-3 Riemann-Hilbert problem and diffraction of a wave by two orthogonal resistive half-planes. (English) Zbl 1243.78023

Summary: Diffraction of a plane electromagnetic wave (\(E\)-polarization) by two orthogonal electrically resistive half-planes is analyzed. The physical problem reduces to a Riemann-Hilbert problem in the real axis for four pairs of analytic functions \(\Phi ^+_j(\eta ) (\eta \in \mathbb C^{+})\) and \(\Phi ^-_j(\eta )= \Phi^+_j(-\eta ) (\eta \in \mathbb C^-)\), \(j=1,2,3,4\), where \(\mathbb C^{+}\) and \(\mathbb C^{-}\) are the upper and lower half-planes. It is shown that the problem is equivalent to two scalar Riemann-Hilbert problems on a plane and a Riemann-Hilbert problem on a genus-3 hyperelliptic surface subject to a certain symmetry condition. A closed-form solution is derived in terms of singular integrals and the genus-3 Riemann theta function.

MSC:

78A45 Diffraction, scattering
30E25 Boundary value problems in the complex plane
35P05 General topics in linear spectral theory for PDEs
45E05 Integral equations with kernels of Cauchy type
35Q15 Riemann-Hilbert problems in context of PDEs
14N25 Varieties of low degree
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