A SYMMETRIC RIEMANN-HILBERT PROBLEM FOR ORDER-4 VECTORS IN DIFFRACTION THEORY

作者：Antipov Y A^*

来源：Quarterly Journal of Mechanics and Applied Mathematics, 2010, 63(3): 349-374.

DOI：10.1093/qjmam/hbq019

A class of n x n matrices of the form G = R(1) + a R(2) (a is a scalar Holder function defined on a contour L, R(1) and R(2) are rational matrices), which admit a closed-form Wiener-Hopf factorization is analyzed. It is shown that the diffraction problem (E-polarization) for a penetrable right-angled wedge with an electrically resistive and a perfectly magnetically conductive sides can be formulated as a Riemann-Hilbert problem for two order-4 vectors with a 4 x 4 matrix coefficient G = R(1) + a R(2). This problem reduces to two scalar Riemann-Hilbert problems and one vector problem for two order-2 vectors with a certain constraint. The order-2 vector problem is solved by employing the theory of the scalar Riemann-Hilbert problem on a genus-1 Riemann surface. The component E(z) of the electric field and the diffraction coefficient are determined.

• 出版日期2010-8