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On Malyshev's method of automorphic functions in diffraction by wedges

Alexander Komech, Anatoli Merzon

Abstract

We describe Malyshev's method of automorphic functions in application to boundary value problems in angles and to diffraction by wedges. We give a consize survey of related results of A. Sommerfeld, S.L. Sobolev, J.B. Keller, G.E. Shilov and others.

On Malyshev's method of automorphic functions in diffraction by wedges

Abstract

We describe Malyshev's method of automorphic functions in application to boundary value problems in angles and to diffraction by wedges. We give a consize survey of related results of A. Sommerfeld, S.L. Sobolev, J.B. Keller, G.E. Shilov and others.
Paper Structure (16 sections, 64 equations, 2 figures)

This paper contains 16 sections, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Diffraction by wedges and radar/sonar detection
  3. Stationary diffraction and boundary value problems in angles
  4. Malyshev's method of automorphic functions
  5. Reduction to undetermined algebraic equation on the Riemann surface
  6. Fourier--Laplace transform
  7. Undetermined algebraic equation on the Riemann surface
  8. Method of automorphic functions
  9. Covering maps
  10. Shift equation
  11. Reduction to the Riemann--Hilbert problem
  12. Nonconvex angles of magnitude $\Phi>\pi$
  13. Time-dependent diffraction by wedge
  14. Limiting amplitude principle
  15. The Sommerfeld diffraction theory and related results
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Incident, reflected, and diffracted waves (the latter in green color).
  • Figure 2: Riemann surface $V\!\!: z_1^2+z_2^2+1=0$ in projection onto the plane ${\rm Im{\space}} z_1,{\rm Im{\space}} z_2$.

Theorems & Definitions (12)

  • Remark 4.1
  • Remark 4.2
  • Remark 4.3
  • Definition 4.4
  • Example 4.5
  • Remark 4.6
  • Remark 4.7
  • Remark 4.8
  • Remark 4.9
  • Remark 7.1
  • ...and 2 more