Schedule for: 24w5506 - Cross-Fertilisation of ideas from the Riemann–Hilbert Technique and the Wiener–Hopf Method

A set dinner is served daily between 5:30pm and 7:30pm in the Xianghu Lake National Tourist Resort.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

By workshop organizers

This talk will offer a review of some of the author’s recent and older studies. In particular, we will focus on pole removal methods for solving a variety of scalar and matrix Wiener-Hopf problems. As time allows we shall also discuss how this method may prove useful to certain Riemann-Hilbert problems.

I will talk about one of the strengths of applying analytic methods in contrast to numerical procedures. In particular, I will examine what information can be extracted from a Wiener--Hopf equation even if the solution cannot be explicitly derived. This will lead to the idea used in diffraction theory called the embedding formula. It provides a way of expressing a solution of a diffraction problem in terms of solutions of similar problems but with different forcings. This allows to reuse solutions and hence reduce computational cost. This is joint work with Andrey Korolkov.

The diffraction of a waveguide mode by a transversal Dirichlet strip in a square lattice waveguide is studied. It is shown that this problem can be formulated as a 4×4 matrix Wiener-Hopf problem, with the kernel reducible to a 2×2 factorable matrix in the symmetric case. The technique used for deriving the Wiener-Hopf equation exploits the symmetric properties of the solution and is validated against the well-known half-plane scattering problem. The study investigates the perspectives of an explicit factorisation of the 2×2 kernel and approximate factorisation using rational approximations, while also discussing potential methods for addressing the non-symmetric case.

Lunch is served daily between 11:30am and 1:30pm in the Xianghu Lake National Tourist Resort

In this talk, I will give an overview of recent developments linking wave theory and multidimensional complex analysis. I will explain how a procedure of complex deformation of the integration surface of Fourier-like highly oscillatory double integrals can lead to closed-form far-field asymptotics results in wave diffraction theory. Each far-field component will be shown to be connected to a special point on the singularity set of the integrand. The procedure will be illustrated through two examples that can be reformulated as two-complex-variables scalar Wiener-Hopf problems: the three-dimensional problem of plane wave diffraction by a quarter-plane and the two-dimensional problem of plane wave diffraction by a penetrable wedge. I will also show how it can be used to shed some light on wave propagation in periodic structures. The talk will cover aspects of several articles written jointly with great collaborators who should be acknowledged: Andrey V. Shanin (Moscow State University), Andrey K. Korolkov (University of Manchester), Valentin D. Kunz (Ohio State University) and I. David Abrahams (University of Cambridge).

Acoustic wave diffraction by a quadrant of sound-soft scatterers Abstract: Motivated by research in metamaterials, we consider the challenging problem of acoustic wave scattering by a doubly periodic quadrant of sound-soft scatterers arranged in a square formation, which we have dubbed the quarter lattice. This leads to a Wiener--Hopf equation in two complex variables with three unknown functions for which we can reduce and solve exactly using a new analytic method. After some suitable truncations, the resulting linear system is inverted using elementary matrix arithmetic and the solution can be numerically computed. This solution is also critically compared to a numerical least squares collocation approach and to our previous method where we decomposed the lattice into semi-infinite rows or columns.

This study introduces an innovative numerical technique utilizing neural networks to factorize Wiener-Hopf matrix kernels through solving Cauchy-Riemann equations, thereby ensuring the kernel's analyticity in both the positive and negative half-planes. To enhance the decomposition process' efficiency, a custom loss function has been developed, incorporating boundary conditions and the inherent symmetry of the matrix. The effectiveness of this method is illustrated through application to the scattering problem involving two parallel plates, showcasing significant computational improvements and enhanced accuracy. The report also discusses strategies for managing exponential terms within kernels, ensuring stability and versatility in various applications. This approach marks an interesting step forward in merging deep learning with complex variable problem solutions. This work has been done by Sicong Liang and Xun Huang.

This talk presents a theoretical framework for assessing sound radiation from a semi-infinite elliptic duct containing a confocal, infinite center-body under uniform subsonic flow conditions. Elliptic ducts are essential components in modern aircraft design, particularly for advanced blended wing body configurations due to their potential for maximizing fuselage pre-compression and improving stealth performance. The proposed model leverages Mathieu functions to describe both the incident and scattered sound fields in elliptic cylindrical coordinates. A comprehensive analytical solution is derived using the Wiener-Hopf technique, yielding accurate near- and far-field predictions. Numerical simulations employing the finite element method are performed to validate the model's accuracy, demonstrating robust agreement with theoretical predictions. Furthermore, this study extends the framework to investigate reflections within the elliptic duct, offering crucial insights into the acoustic properties of these structures undergoing flow conditions - relevant for noise control and optimization of turbofan engine inlets and blended wing body applications. This is joint work with Xun Huang.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

We consider a steady state 2d problem of a semi-infinite crack propagating at subsonic speed along the interface between two inhomogeneous half-planes. It is assumed that the Lame coefficients and density are power functions of depth. By means of the Fourier and Mellin transforms the physical model reduces to scalar Riemann-Hilbert problems with discontinuous coefficients. The Riemann-Hilbert problem coefficients are unknown and constitute the solution of a vector Carleman problem for two functions in a strip with two shifts. It is shown that the latter problem is equivalent to a system of singular integral equations with a fixed singularity. A numerical recipe for its solution and the behavior of the displacements and stresses at the crack tip are discussed.

In this talk, we will outline how the problem of studying the diffraction coefficient appearing in the context of diffraction of a monochromatic plane-wave by a quarter-plane can be reduced to studying the monodromy of some one-complex-variable spectral functions. Roughly speaking, studying such monodromy requires analysis of how the 'physical' 2-sphere (within $\mathbb{R}^3$), which is linked to the quarter-plane problem via separation of variables, interacts with a 'sphere at infinity'. We will explain the ideas of how the sought-after monodromy can be found, and how our techniques can be applied to the quarter-plane problem. Time permitting, we will discuss possible adaptations to other diffraction problems.

Wiener-Hopf factorization (WHf) theory encompasses several important results in probability and stochastic processes, as well as in operator theory. The importance of the WHf stems not only from its theoretical appeal, manifested, in part, through probabilistic interpretation of analytical results, but also from its practical applications in a wide range of fields, such as fluctuation theory, insurance, and finance. The various existing forms of WHf for Markov chains, strong Markov processes, Levy processes, and Markov additive process, have been obtained only in the time-homogeneous case. However, there are abundant real life dynamical systems that are modeled in terms of time-inhomogenous processes, and yet the corresponding WHf theory is not available for this important class of models. In this talk, I will discuss our recent works on WHf for time-inhomogensous Markov chains and time-inhomogenous diffusion processes. To the best of our knowledge, these studies are the first two attempts to investigate the WHf for time-inhomogeneous Markov processes. This talk is based on joint works with Tomasz R. Bielecki, Ziteng Cheng, and Igor Cialenco.

This work is a natural continuation of our earlier paper [1]. Using the Janashia-Lagvilava method [2,3], we approximate an NxN matrix function G, which has a factorable principle minors, arbitrarily close with matrix function G0 whose explicit factorization is reduced to the factorization of polynomial matrices with monomial determinant tn. Then we propose a new algorithm for factorization of such polynomial matrices. In the end, we arrive to the Wiener-Hopf factorization of G0 . The work was supported by the EU through the H2020-MSCA-RISE-2020 project EffectFact, Grant agreement ID: 101008140. References [1] L. Ephremidze and I. Spitkovsky, On explicit Wiener-Hopf factorization of 2x2 matrices in a vicinity of a given matrix, 2020, Proceedings of the Royal Society, A 476: 20200027. https://doi.org/10.1098/rspa.2020.0027 [2] G. Janashia, E. Lagvilava, and L. Ephremidze, A new method of matrix spectral factorization, IEEE Trans. Inform. Theory, 57 (2011), 2318-2326. DOI 10.1109/TIT.2011.2112233 [3] L. Ephremidze, F. Saied, and I. Spitkovsky, On the algorithmization of Janashia-Lagvilava matrix spectral factorization method, IEEE Trans. Inform. Theory, 64 (2018), 728-737. DOI: 10.1109/TIT.2017.2772877

The general theme of the talk are representations of functions on a curve by analytic functions in a neighborhood of that curve. Classical examples are the Riesz decomposition and the (generalized) Wiener-Hopf factorization. These are closely linked with boundary value problems and transmission problems of Riemann-Hilbert type. In the first part we explore some original sources, in particular Riemann's thesis, which contains fundamental ideas and a new paradigm for investigating complex functions. The second part is devoted to a hyperbolic version of the Riesz decomposition, the Blaschke representation of functions with sup-norm not exceeding 1. Originally it was established (in weaker form) in the context of Wiener-Hopf methods in scattering theory. We demonstrate how the result can be generalized using the interplay with nonlinear Riemann-Hilbert problems. The third part has more entertaining character: we interpret the material in the context of hyperfunctions and Plato's Cave Allegory, illustrate it by phase plots, and reveal the meaning of the title.

A comprehensive theory to study electromagnetic (EM) wedge diffraction problems immersed in complex media is introduced in spectral domain in the framework of Wiener-Hopf technique. This innovative and effective spectral theory has its foundations on: 1) a generalization of transverse equations theory applied to angular regions, 2) the characteristic Green function procedure, 3) the Wiener–Hopf method, and 4) a novel strategy for solving more generalized Wiener–Hopf equations (GWHEs) based on a generalization of Fredhom factorization. The methodology has demonstrated its efficacy in analyzing problems with wedges immersed in isotropic media. With the present work we extend the theory and applications to arbitrary linear EM media. It is important to note that spectral techniques, including the Sommerfeld–Malyuzhinets (SM) method, the Kontorovich–Lebedev (KL) transform method, and the Wiener–Hopf (WH) method, are well established, fundamental, and effective instruments for the accurate analysis of diffraction problems constituted by wedges in media with a single propagation constant. Here we extend the WH technique to the analysis of wedge diffraction in media with multiple propagation constants. To our knowledge, the proposed mathematical approach represents the first extension of spectral analysis of wedge diffraction problems immersed in complex arbitrary linear media. Validation through fundamental examples is proposed. The method is applicable to other physics beyond electromagnetics. This is joint work with Vito Daniele. 1. A.D. Bresler, N. Marcuvitz, “Operator Methods in Electromagnetic Field Theory,” Report R-495,56, PIB425; MRI Polytechnic Institute of Brooklyn: New York, NY, USA, 1956 2. L.B. Felsen, N. Marcuvitz, Radiation and Scattering of Waves; Prentice-Hall: Englewood Cliffs, NJ, USA, 1973 3. V. Daniele, G. Lombardi, “The Generalized Wiener-Hopf Equations for wave motion in angular regions: electromagnetic application,” Proc. R. Soc. A, 477:20210040, n.2252, pp. 1-27, 2021 4. V. Daniele, G. Lombardi, Scattering and Diffraction by Wedges 1: The Wiener-Hopf Solution—Theory, Hoboken, NJ: John Wiley & Sons, Inc. London, UK: ISTE, 2020 5. V. Daniele, G. Lombardi, Scattering and Diffraction by Wedges 2: The Wiener-Hopf Solution - Advanced Applications, Hoboken, NJ: John Wiley & Sons, Inc. London, UK: ISTE, 2020 6. V. Daniele, G. Lombardi, “Spectral Analysis of Electromagnetic Diffraction Phenomena in Angular Regions Filled by Arbitrary Linear Media,” Applied Science, 14, no. 19: 8685, pp. 1-37, 2024 7. V. Daniele, G. Lombardi, “The generalized Wiener–Hopf equations for the elastic wave motion in angular regions,” Proc. R. Soc. A, 478:20210624, n. 2257, pp.1-29, 2022

In boundary value problems, one key advantage of analytical solutions over numerical ones is that the dependence on parameters is explicit. This explicit dependence is particularly useful for optimizing regimes and addressing inverse problems. Here, we explore an intermediate method known as embedding, where complete analytical solutions are not derived, yet it is still possible to obtain some analytical dependence of the solution on specific parameters. A systematic approach to deriving embedding formulas for plane wave diffraction problems was outlined in [1]. The steps involve: identifying an operator H that “annihilates” the incident plane wave, creates singularities near the edges of the scatterer, preserves boundary conditions, and also commutes with the Laplace operator; introducing the auxiliary oversingular solutions; studying the combination H[u] + Kivi , where Ki are some constants, and vi are auxiliary solutions. It is then shown, through uniqueness arguments, that the solution to the plane diffraction problem can be expressed in terms of these auxiliary solutions. In this work, we propose an alternative approach. Simply put, we argue that every embedding formula is rooted in a matrix Wiener–Hopf equation, and the embedding formula is essentially the canonical solution [2] to this matrix Wiener–Hopf problem. We demonstrate the effectiveness of this approach by revisiting well-known problems, such as the problem of diffraction by a strip, and the problem of diffraction by a wedge. Additionally, we derive new embedding formulas for problems in discrete diffraction theory. References [1] R. V. Craster, A. V. Shanin, and E. M. Doubravsky. Embedding formulae in diffraction theory. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 459(2038):2475–2496, Oct. 2003. [2] F. D. Gakhov. Riemann’s boundary problem for a system of n pairs of functions. Uspekhi matematicheskikh nauk, 7(4):3–54, 1952.

The problems in Fracture Mechanics and Mechanics of Composites are often formulated in terms of jumps in elastic fields across some surfaces or curves. In the talk, I will discuss the approach that is based on the use of the theories of elastic potentials for solving such problems. Both two- and three-dimensional problems will be discussed. It will be demonstrated that, in two dimensional settings, it is beneficial to use the complex variable forms of the potentials. Several classes of problems that have important applications in Fracture Mechanics and Mechanics of Composites will be considered to illustrate the effectiveness and robustness of the approach.

The Wiener-Hopf Technique is used to solve two mixed-boundary problems in aeroacoustics. In the first problem, an analytical Green’s function for the serrated edge wave scattering problem is solved using the Wiener-Hopf technique. A closed-form analytical Green’s function is obtained for piecewise linear serrations and compared with the canonical Green’s function for straight edges. The analytical Green’s function is verified using the finite element method. Both noise reduction spectra and directivity patterns are studied as a function of source position. Physical mechanism of sound reduction is discussed. In the second problem, the generation of instability waves in a supersonic jet induced by acoustic wave impingement is examined. To obtain the newly-excited instability wave, the scattered sound field due to the acoustic impingement is first solved using the Weiner-Hopf technique, with the kernel function factored using asymptotic expansions and overlapping approximations. Subsequently, the unsteady Kutta condition is imposed at the nozzle lip, enabling the derivation of the dispersion relation for the newly-excited instability wave. A linear transfer function between the upstream forcing and the newly- excited instability wave is obtained. The amplitude and phase delay and their dependence on the frequency are examined. The new model shows improved agreement between the predicted screech frequencies and the experimental data compared to classical models.

The Unified Transform Method is a technique for analyzing boundary value problems for linear and integrable nonlinear PDEs. This talk will present a new transform method for Laplace's equation in non-circular and non-convex planar domains. This new transform pair uses the Szego Kernel as an alternative for the Cauchy kernel. This is join work with L. Lanazani, S. Llewellyn Smith, and E. Luca

Hydroelasticity describes any problem in which fluid and elastic structures interact and has found a wide and ever-growing range of application. This is especially true in the field of marine engineering and polar geophysics, but it has also recently become a significant area of research in wave energy conversion. As always, simple analytic solutions have proven to be of great use to understand complex behaviour such as is exhibited in hydroelastic systems. I will present here some examples involving floating thin plates in which we have been able to find analytic solutions using methods from complex variables and showcase some of the applications.

We consider anti-plane waves propagating in a structured quadrant formed from periodically placed point masses interconnected by springs [1]. The waves are assumed to be produced by the action of a point force located along the lateral boundary of the quadrant. We employ the Fourier transforms with respect to the principal lattice coordinates to reduce the governing equations to a functional equation with 3 unknowns. The method of [2] can then be applied to obtain a Wiener-Hopf equation for one of the unknowns. This equation can then be solved in a standard way and the corresponding solution can be used to determine the remaining unknowns. In particular, we discuss wave generation and scattering within a quadrant having (a) a fully unconstrained boundary and (b) a constrained vertical boundary. The analytical results are also accompanied by numerical illustrations demonstrating the response of the lattice at various frequencies. The talk is based on joint work with Anastasia Kisil and Gennady Mishuris. References: [1] Nieves M, Kisil A, Mishuris GS. 2024 Analytical and numerical study of anti-plane elastic wave scattering in a structured quadrant subjected to a boundary point load. Proc. R. Soc. A 480: 20240099. https://doi.org/10.1098/rspa.2024.0099 [doi.org]

Yi-Ze Wang*1,2, Kuan-Xin Huang3 and Yue-Sheng Wang1,2,3 1 Department of Mechanics, Tianjin University, Tianjin 300350, China 2National Key Laboratory of Vehicle Power System, Tianjin 300350, China 3Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing 100044, China *Corresponding author: E-mail: wangyize@tju.edu.cn Abstract: In the last decade, elastic wave metamaterials have received considerable attention due to the prospective application to elastic and acoustic wave propagations. Then, concerns about their fracture and damage issues have risen. On the other hand, Slepyan and his colleagues have made remarkable contributions to dynamic fracture of discrete structural media with Wiener-Hopf technique. In the present work, our recent works about the dynamic fracture of locally resonant elastic wave metamaterials are reported. Based on Wiener-Hopf method and approach developed by Slepyan, three typical problems and their progress are shown. The results show that the local resonators can increase the splitting resistance and restrain the crack kinking by absorbing the fracture energy. This topic is expected to be helpful to enhance the strength and safety of advanced materials and structures. Acknowledgements: The authors wish to express gratitude for the support provided by the National Natural Science Foundation of China (Grant Nos. 12021002, 11991031 and 12425203).

The purpose of the presentation is to discuss the convolution integro-differential equations on Lie groups and their applications to some equations of Mathematical Physics. In this framework we suggest to underline the role of Generic Bessel potential spaces (GBPS) to the structure of underlying Lie group. Definition of GBPS are based on generic differential operators from the Lie algebra of the Lie group. Such generic Bessel potential spaces are adapted better to the investigation of integro-differential (of pseudo-differential) operators on Lie groups. We concentrate investigation on a Lie groups $\{G,x\circ y\}$ with the group operation $x\circ y$, which are homeomorphic to the Lie group $\{\mathbb{R}^n,x\circ y=x+y\}$. Then on $\{G,x\circ y\}$ we have uniquelly defined Haar measure $d_G\mu$, the Fourier transform $\mathcal{F}_G$, its inverse $\mathcal{F}^{-1}_G$ and generic differential operators $\mathfrak{D}_1,\ldots,\mathfrak{D}_n$, generated by the vector fields from the corresponding Lie algebra. The dual group is then $\widehat{G}=\mathbb{R}^n$ and Convolution operators are % EQUATION $$\begin{eqnarray}\label{e1} \boldsymbol{W}^0_{a,G}:=\mathcal{F}^{-1}_G a\mathcal{F}_G\;:\;\mathbb{S}(G)\to\mathbb{S}'(G), \end{eqnarray}$$ where the symbol $a(\xi)$ is a distribution on the dual group $a\in\mathbb{S}'(\widehat{G})=\mathbb{S}'(\widehat{\mathbb{R}^n})$, $\mathbb{S}(G)$ is the Schwartz spaces of fast decaying smooth functions and $\mathbb{S}'(G)$ is the spaces of distributions. We will expose several examples of Lie groups and corresponding GBPS. Then we concentrate on the investigation of boundary value problems (BVPs) for the Laplace-Beltrami equation on a hypersurface $\mathcal{C}$ with the Lipschitz boundary, containing a finite number of angular points (knots). The Dirichlet, Neumann and mixed type BVPs are considered in two different non-classical setting: A) Solutions are sought in the classical Bessel potential spaces $\mathbb{H}^s_p(\mathcal{C}$), $11/p$; A) Solutions are sought in the generic Bessel potential spaces with weight$\mathbb{G}\mathbb{H}^s_p(\mathcal{C,\rho}$). By the localization the problem is reduced to the investigation of Model Dirichlet, Neumann and mixed BVPs for the Laplace equation in a planar angular domains, also in cases of double angles. Explicit criteria for the Fredholm property and the unique solvability of the initial BVPs in both cases are obtained and, for the Generic Bessel potential spaces also singularities of solutions at knots of the mentioned BVPs are indicated explicitely. The first part part of the presentation is based on joint results with M. Ruzhanski, D, Cardona, A. Hendrix (Ghent) and the second part-on joint work with M. Caava (Tbilisi).

Matrix Klein-Gordon equation (MKGE) [1,2] is an efficient instrument for description of field in waveguides of various nature. This equation contains matrix coefficients describing the waveguide’s structure, partial derivatives over the longitudinal coordinate and partial derivatives over time . is a vector of unknowns, it describes field in the cross-section. There is an excitation in the right-hand side. MKGE may be obtained in the result of discretisation of the waveguide’s cross-section, while its longitudinal coordinate remains continuous [3]. The state vector may be composed of nodal values of some field variable. Here we discuss meaning of each term in MKGE. Partially, we show that MKGE described flexural waves correctly, even though the flexural waves are described with an equation containing the term with . Finally, we show an example from industry [4]. This is joint work with A.V. Shanin, , A.I. Korolkov and E.L. Shelest. [1] Finnveden, S., Evaluation of modal density and group velocity by a finite element method. Journal of Sound and Vibration, 273(1-2), (2004) 51-75. [2] Shanin, A. V., Korolkov, A. I., & Kniazeva, K. S., Saddle point method for transient processes in waveguides. Journal of Theoretical and Computational Acoustics, 30(04), (2022) 2150018. [3] Aalami, B., Waves in prismatic guides of arbitrary cross section, Journal of Applied Mechanics, 40 (1973) 1067-1072. [4] Kniazeva, K. S., Saito, Y., Korolkov, A. I., & Shanin, A. V., Saddle Point Method

A 2D Fourier integral is studied, whose transformant is a function having algebraic growth at infinity and holomorphic everywhere except some polar and branching sets of complex codimension 1. The integration surface is the real plane, possibly slightly shifted to avoid the singularities of the transformant. The aim of the talk is to build the deformation of the integration surface into some other surface to make the exponential factor of the Fourier integral decaying as fast as possible on it. According to the multidimensional Cauchy's theorem, a deformation (homotopy) not hitting the singular sets does not change the value of the integral. At the same time, a proper deformation makes the integral more suitable for numerical evaluation or for asymptotical investigation. A general procedure of deformation of the integration surface is proposed in the talk. The resulting surface is a sum of components stemming from the special points of the singularities of the transformant, such as the saddle on singularities or the crossings (see [1]). The components have topological structure dictated by the special point type. Thus, the field becomes splitted explicitly into terms corresponding to the special points of the singularities. The work is being done in collaboration with Raphael C. Assier and Andrey I. Korolkov from the University of Manchester.

In this talk, we study a problem for a nano-sized material surface attached to the boundary of an elastic isotropic semi-plane. The material surface is modeled using the Steigmann-Ogden form of surface energy. The study of stationary points of the total elastic energy functional produces a boundary-value problem with non-classical boundary conditions. This problem is solved by using complex analysis methods. With the help of Cauchy-type integral representations of stresses and displacements, the problem can be reduced to either a system of two singular integral equations or a single singular integral equation. The numerical solution of the system of singular integral equations is obtained by expanding each unknown function into a series based on Chebyshev polynomials. The accuracy of the numerical procedure is studied, and various numerical examples for different values of the surface energy parameters are considered.

In this talk, we exploit the functionality of the ExactMPF package to address the general factorization problem of determining whether a given strictly nonsingular 2 × 2 matrix function admits canonical or stable factorization. The idea is to approximate the latter by a sequence of polynomial matrix functions that admit exact factorization while preserving the same properties as the original matrix function. To achieve this goal, we propose an effective sufficient criterion that guarantees that, starting from some element, the given matrix function belongs to a small neighborhood of the stability domain of each subsequent element of the sequence. The theoretical results supporting the method rely on an appropriate normalization of the approximate matrix functions. Additionally, we present some numerical results highlighting the proposed procedure. (Co-Authors: N. Adukova, V. Adukov)