Vector Riemann–Hilbert problem with almost periodic and meromorphic coefficients and applications

Abstract

The vector Riemann-Hilbert problem is analysed when the entries of its matrix coefficient are meromorphic and almost periodic functions. Three cases for the meromorphic functions, when they have (i) a finite number of poles and zeros (rational functions), (ii) periodic poles and zeros, and (iii) an infinite number of non-periodic zeros and poles, are considered. The first case is illustrated by the heat equation for a composite rod with a finite number of discontinuities and a system of convolution equations; both problems are solved explicitly. In the second case, a Wiener-Hopf factorization is found in terms of the hypergeometric functions, and the exact solution of a mixed boundary value problem for the Laplace equation in a wedge is derived. In the last case, the Riemann-Hilbert problem reduces to an infinite system of linear algebraic equations with the exponential rate of convergence. As an example, the Neumann boundary value problem for the Helmholtz equation in a strip with a slit is analysed.

Quiver CFT at strong coupling

“…Matrix factorization is a substantially more complicated problem (see [25] for a review) for which there is no simple plug-in solution. Fortunately, for the particular case of (4.16) the Riemann-Hilbert factorization has been carried out explicitly [26]. The Wiener-Hopf factors were found in [26] by exploiting analytic properties of the hypergeometric functions and linear identities among them.…”

“…Fortunately, for the particular case of (4.16) the Riemann-Hilbert factorization has been carried out explicitly [26]. The Wiener-Hopf factors were found in [26] by exploiting analytic properties of the hypergeometric functions and linear identities among them. In principle, an explicit formula is all we need, but we would like to present a derivation that highlights connections to the inverse scattering problem.…”

An Iterative Wiener--Hopf Method for Triangular Matrix Functions with Exponential Factors

“…The resulting Wiener-Hopf equation is of a more general type than (1). It has been shown [5] that the solution could be reduced to finding two constants C 1 and C 2 by the Wiener-Hopf method. In this example it is possible to obtain the exact solution, which provides a good way to test out the ideas that are introduced in this paper.…”

“…Wiener-Hopf equations of the type (1) have been the topic of previous research, for example, in the case of meromorphic matrix entries [4,5] and in the framework of almost periodic functions [20]. However, presently no complete solution of (1) is known.…”

Aerodynamic noise from rigid trailing edges with finite porous extensions

“…The remaining functions A(α), B(α) and C(α) are known and L is a positive constant. Many problems have a Wiener-Hopf equation of this type (Antipov 2015;Aktosun et al 1992) or could be reduced to such equation by performing matrix manipulations. Presently no complete solution of (1.1) is known.…”