Matrix Wiener-Hopf Factorisation

Abstract

A direct method is described for effecting the explicit Wiener-Hopf factorisation of a class of (2 x 2)-matrices. The class is determined such that the factorisation problem can be reduced to a matrix Hilbert problem which involves an upper or lower triangular matrix. Then the matrix Hilbert problem can be further reduced to three scalar Hilbert problems on a half-line, which are solvable in the standard manner. The factorisation technique is applied to the matrices that arise from two problems in diffraction theory, thus permitting these diffraction problems to be solved in closed form (at least in principle).

A brief historical perspective of the Wiener–Hopf technique

“…Matrix Wiener-Hopf kernels are fundamentally distinct from their scalar counterparts in that there is no algorithmic approach to determining the factorization (4) of the transformed kernel [16]. Exact factorization can be achieved for matrices with certain special features: those that are upper (or lower) triangular in form; those that are of Khrapkov-Daniele, i.e., commutative, form (see [17][18][19][20]); those whose elements comprise meromorphic functions [21,22]; kernels with special singularity structure that allows the Wiener-Hopf equation to be recast into uncoupled Riemann-Hilbert problems [23][24][25]; and N × N matrices with special algebraic or group structure [26][27][28]. For more details on exact matrix kernel factorization the interested reader is referred to the last mentioned article and to references cited in [29].…”

Diffraction of Plane Waves by a Slit in an Infinite Soft-Hard Plane

“…There is, as yet, no general procedure of factorization of such matrices, although the factorization for a restricted class of matrices has been achieved. For example the Wiener-Hopf-Hilbert method, introduced by Hurd [25], Rawlins [26] and Rawlins and Williams [27], is a powerful tool in the case when the kernel matrix contains branch-point singularities, while the Daniele-Kharapkov method, proposed independently by Daniele [28] and Kharapkov [29], is effective for the class of matrices having only pole-singularties and branch-point singularities. Another detailed survey for the matrix factorization methods with reference to applications of these methods to different diffraction problems may also be found in a paper by Büyükaksoy et al [30].…”

Multiple diffraction of plane waves by a soft/hard strip

“…It can be seen that this approach includes also the case stated by Rawlins and Williams [6]. A more general procedure which is applicable to the cases for which the kernel matrix has only poles or poles and branch-point singularities is given by Khrapkov [7], Daniele [8], Rawlins [9] and Jones [10].…”