A high order continuation method to locate exceptional points and to compute Puiseux series with applications to acoustic waveguides

Abstract

A numerical algorithm is proposed to explore in a systematic way the trajectories of the eigenvalues of non-Hermitian matrices in the parametric space and exploit this in order to find the locations of defective eigenvalues in the complex plane. These non-Hermitian degeneracies also called exceptional points (EP) have raised considerable attention in the scientific community as these can have a great impact in a variety of physical problems.The method requires the computation of successive derivatives of two selected eigenvalues with respect to the parameter so that, after recombination, regular functions can be constructed. This algebraic manipulation permits the localization of exceptional points (EP), using standard root-finding algorithms and the computation of the associated Puiseux series up to an arbitrary order. This representation, which is associated with the topological structure of Riemann surfaces allows to efficiently approximate the selected pair in a certain neighbourhood of the EP.Practical applications dealing with guided acoustic waves propagating in straight ducts with absorbing walls and in periodic guiding structures are given to illustrate the versatility of the proposed method and its ability to handle large size matrices arising from finite element discretization techniques. The fact that EPs are associated with optimal dissipative treatments in the sense that they should provide best modal attenuation is also discussed.

Beyond the limitations of perturbation methods for real random eigenvalue problems using Exceptional Points and analytic continuation

“…Recently an algorithm to precisely locate EP has been proposed by Nennig and Perrey-Debain [36]. It exploits high order eigenvalue derivatives and analytic continuation of the parameter in the vicinity of the EP.…”

“…The algorithm proposed by Nennig and Perrey-Debain [36] also exploits the high order derivatives of the veering pair of eigenvalues. The key idea is to introduced two analytic auxiliary functions of the eigenvalues pair…”

“…[28,49] but they are limited to first or second terms and the computation performed at EP are hard to handle numerically. The advantage of the method proposed in [36] is to compute eigenpairs and derivatives far from the singularity where the standard eigenvalue solver may encounter some difficulties.…”

“…The key idea of this paper is to use non-Hermitian framework and analytic continuation around the EP, introduced in [36], to go beyond the standard limitations of perturbation methods. For the sake of simplicity, a single varying parameter is considered, but the results could be extended to multi-parametric systems.…”

“…See for instance Murthy's survey [40] dedicated to complex non-Hermitian matrices. In the context of this work, the direct method proposed by Andrew et al [39] based on the bordered matrix, is adopted as in [36]. It is noteworthy that this method can be used to obtained the eigenvalue derivatives with respect to several variables [39].…”