Komech A., Merzon A. Stationary Diffraction by Wedges: Method of Automorphic Functions on Complex Characteristics

Springer, 2019. — 157 p. — (Lecture Notes in Mathematics 2249). — ISBN: 978-3-030-26698-1.This book presents a new and original method for the solution of boundary value problems in angles for second-order elliptic equations with constant coefficients and arbitrary boundary operators. This method turns out to be applicable to many different areas of mathematical physics, in particular to diffraction problems in angles and to the study of trapped modes on a sloping beach.
Giving the reader the opportunity to master the techniques of the modern theory of diffraction, the book introduces methods of distributions, complex Fourier transforms, pseudo-differential operators, Riemann surfaces, automorphic functions, and the Riemann–Hilbert problem.
The book will be useful for students, postgraduates and specialists interested in the application of modern mathematics to wave propagation and diffraction problems.The Early Theory of Diffraction
Fresnel–Kirchhoff Diffraction Theory
Stationary and Time-Dependent Diffraction
The Sommerfeld Theory of Diffraction by Half-Plane
Diffraction by Wedge After Sommerfeld’s Article
Stationary Boundary Value Problems in Convex Angles
Extension to the Plane
Boundary Conditions via the Cauchy Data
Connection Equation on the Riemann Surface
On Equivalence of the Reduction
Undetermined Algebraic Equations on the Riemann Surface
Automorphic Functions on the Riemann Surface
Functional Equation with a Shift
Lifting to the Universal Covering
The Riemann-Hilbert Problem on the Riemann Surface
The Factorization
The Saltus Problem and Final Formula
The Reconstruction of Solution and the Fredholmness
Extension of the Method to Non-convex Angle
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