Searcharxiv⌕ Search

arXiv subjects

Habib Ammari

Publications and source records attributed to Habib Ammari.

At least 37 records · Page 2Linked to original sources
arXiv2026

Reduced Order Model for Broadband Superabsorption of Waves by Metascreens

This work presents a new design for broadband absorption of low-frequency acoustic waves using a thin coating made of subwavelength acoustic resonators arranged periodically on a reflective surface. We first study the associated scattering problem and the corresponding subwavelength resonance problem, and then derive analytical approximations for the resonant frequencies and the reflection coefficient in terms of the periodic capacitance matrix in a half-space with a Dirichlet boundary condition. These approximations yield an effective macroscopic description of the coating via an impedance boundary condition and clarify the mechanism of superabsorption through an approximate coupling condition. Moreover, they lead to a reduced order model that enables efficient evaluation of the scattered waves over a frequency band and accelerates broadband absorption design. Building on this reduced order model, we develop a gradient based shape optimization method using shape derivatives of the resonant quantities to achieve broadband absorption. Numerical experiments demonstrate the broadband performance and the effectiveness of the proposed design procedure.

math.NA↗
arXiv2026

Non-Hermitian Fabry-Pérot Resonances

We characterise non-Hermitian Fabry-Pérot resonances in high-contrast resonator systems and study the properties of their associated resonant modes from continuous differential models. We consider two non-Hermitian effects: the exceptional point degeneracy and the skin effect induced by imaginary gauge potentials. Using the propagation matrix formalism, we characterise these two non-Hermitian effects beyond the subwavelength regime. This analysis allows us to (i) establish the existence of exceptional points purely from radiation conditions and to (ii) prove that the non-Hermitian skin effect applies uniformly across resonant modes, yielding broadband edge localisation.

math-ph↗
arXiv2026

A Real-Space Formulation of the Zak Phase via Weyl m-Functions

We establish a new, real-space formula for the Zak phase for one dimensional periodic Jacobi operators in terms of the Weyl $m_+$-function that does not rely on Floquet-Bloch theory. This novel representation highlights the dependence of the Zak phase on boundary terms. Moreover, we show how to recover the classical quantisation of the Zak phase for periodic Jacobi operators with inversion symmetric fundamental cells.

math-ph↗
arXiv2026

A Tight-binding Approach for Computing Subwavelength Guided Modes in Crystals with Line Defects

In this paper, we develop an accurate and efficient framework for computing subwavelength guided modes in high-contrast periodic media with line defects, based on a tight-binding approximation. The physical problem is formulated as an eigenvalue problem for the Helmholtz equation with high-contrast parameters. By employing layer potential theory on unbounded domains, we characterize the subwavelength frequencies via the quasi-periodic capacitance matrix. Our main contribution is the proof of exponential decay of the off-diagonal elements of the associated full and quasi-periodic capacitance matrices. These decay properties provide error bounds for the banded approximation of the capacitance matrices, thereby enabling a tight-binding approach for computing the spectral properties of subwavelength resonators with non-compact defects. Various numerical experiments are presented to validate the theoretical results, including applications to topological interface modes.

math-ph↗
arXiv2025

Topological interface modes in aperiodic subwavelength resonator chains

We consider interface modes in block disordered subwavelength resonator chains in one dimension. Based on the capacitance operator formulation, which provides a first-order approximation of the spectral properties of dimer-type block resonator systems in the subwavelength regime, we show that a two-fold topological characterization of a block disordered resonator chain is available if it is of dominated type. The topological index used for the characterization is a generalization of the Zak phase associated with one-dimensional chiral-symmetric Hamiltonians. As a manifestation of the bulk-edge correspondence principle, we prove that a localized interface mode occurs whenever the system consists of two semi-infinite chains with different topological characters. We also illustrate our results from a dynamic perspective, which provides an explicit geometric picture of the interface modes, and finally present a variety of numerical results to complement the theoretical results.

physics.optics↗
arXiv2025

Analysis of nonlinear resonances in resonator crystals: Tight-binding approximation and existence of subwavelength soliton-like solutions

This work provides a mathematical framework for elucidating physical mechanisms for confining waves at subwavelength scales in periodic systems of nonlinear resonators. A discrete approximation in terms of the linear capacitance operator is provided to characterize the nonlinear subwavelength resonances. Moreover, the existence of subwavelength soliton-like localized waves in periodic systems of nonlinear resonators is proven. As a by-product, a tight-binding approximation of the capacitance operator is shown to be valid for crystals of subwavelength resonators. Both full- and half-space crystals are considered. The framework developed in this work opens the door to the study of topological properties of periodic lattices of subwavelength nonlinear resonators, such as the emergence of nonlinearity-induced topological edge states, and to elucidate the interplay between nonlinearity and disorder.

math-ph↗
arXiv2025

Competing edge and bulk localisation in non-reciprocal disordered systems

We investigate the competing mechanisms of localisation in one-dimensional block disordered subwavelength resonator systems subject to non-reciprocal damping, induced by an imaginary gauge potential. Using a symmetrisation approach to enable the adaptation of tools from Hermitian systems, we derive the limiting spectral distribution of these systems as the number of blocks goes to infinity and characterise their spectral properties in terms of the spectral properties of their constituent blocks. By employing a transfer matrix approach, we then clarify, in terms of Lyapunov exponents, the competition between the edge localisation due to imaginary gauge potentials and the bulk localisation due to disorder. In particular, we demonstrate how the disorder acts as insulation against the non-Hermitian skin effect, preventing edge localisation for small imaginary gauge potentials.

math-ph↗
arXiv2025

Dielectric scattering resonances for high-refractive resonators with cubic nonlinearity

This work establishes a rigorous mathematical framework for the analysis of nonlinear dielectric resonances in wave scattering by high-index resonators with Kerr-type nonlinearities. We consider both two- and three-dimensional settings and prove the existence of nonlinear dielectric resonances in the subwavelength regime, bifurcating from the zero solution at the corresponding linear resonances. Furthermore, we derive asymptotic expansions for the nonlinear resonances and states in terms of the high contrast parameter $τ$ and the normalization constant. For a symmetric dimer of resonators, these small-amplitude nonlinear resonant states exhibit either symmetric or antisymmetric profiles. In three dimensions, under conditions valid in the dilute regime, we prove that as the field amplitude increases, mode hybridization induces a symmetry-breaking bifurcation along the principal symmetric solution branch at a critical amplitude. This bifurcation gives rise to two asymmetric resonant states, each localized on one of the particles in the dimer. Remarkably, in two dimensions, we show that no such symmetry-breaking bifurcation exists along the principal solution branches, owing to the distinct scaling behavior of the principal nonlinear subwavelength resonance arising from the logarithmic singularity.

math.AP↗
arXiv2025

Floquet$-$Bloch analysis of wave propagation with time-periodic coefficients

This paper presents a numerical investigation of acoustic wave propagation in an obstacle with periodically time-modulated material parameters. We focus on the numerical construction of Floquet$-$Bloch solutions, which are quasi-periodic kernel elements of the hyperbolic operator appearing on the left-hand side of the acoustic wave equation. Using the temporal Fourier expansion yields a system of coupled harmonics, which can be truncated. Rewriting this system then provides different (generally nonlinear) eigenvalue formulations for discretized Floquet$-$Bloch solutions. Deriving energy estimates and the necessary conditions for Riesz$-$Schauder theory show basic properties of the occurring Floquet exponents. To derive fully discrete schemes, we employ a general Galerkin space discretization. Under assumptions on the relation of the temporal Fourier truncation and the Galerkin space discretization, we prove that the approximated Floquet exponents exhibit the same limitations as their continuous counterparts. Moreover, the approximated modes are shown to satisfy the defining properties of Floquet$-$Bloch solutions, with a defect that tends to zero as the number of harmonics approaches infinity. Numerical experiments demonstrate the effectiveness of the proposed approach and illustrate the theoretical findings.

math.NA↗
arXiv2025

Effective Medium Theory for Time-modulated Subwavelength Resonators

This paper provides a general framework for deriving effective material properties of one-dimensional, time-modulated systems of subwavelength resonators. It applies to subwavelength resonator systems with a general form of time-dependent parameters. We show that the resonators can be accurately described by a point-scattering formulation when the width of the resonators is small. In contrast to the static setting, where this point interaction approximation yields a Lippmann-Schwinger equation for the effective material properties, the mode coupling in the time-modulated case instead yields an infinite linear system of Lippmann-Schwinger-type equations. The effective equations can equivalently be written as a system of differential equations. Moreover, we introduce a numerical scheme to approximately solve the system of coupled equations and illustrate the validity of the effective equation.

physics.optics↗
arXiv2025

Subwavelength Localisation in Disordered Systems

We elucidate the different mechanisms of wave localisation in disordered finite systems of subwavelength resonators, where the disorder is in the spatial arrangement of the resonators. To do so, we employ the capacitance matrix formalism and develop a variety of tools to understand localisation in this setting. Namely, we adapt the Thouless criterion of localisation to quantify the various mechanisms of localisation. We also employ a propagation matrix approach to prove a Saxon-Hutner type theorem ensuring the existence of band gaps for disordered systems and to predict the corresponding Lyapunov exponents. We further investigate the stabilities of bandgaps and localised eigenmodes with respect to fluctuations in the system by introducing a method of defect mode prediction based on the discrete Green function and illuminating the phenomena of Anderson localisation and level repulsion under global disorder.

cond-mat.dis-nn↗
arXiv2025

Mathematical Foundation of Sparsity-based Multi-snapshot Spectral Estimation

In this paper, we study the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements in a bounded domain. We aim to provide a mathematical foundation for sparsity-based super-resolution in such spectral estimation problems in both one- and multi-dimensional spaces. In particular, we estimate the resolution and stability of the location recovery when considering the sparsest solution under the measurement constraint, and characterize their dependence on the cut-off frequency, the noise level, the sparsity of point sources, and the incoherence of the amplitude vectors of point sources. Our estimate emphasizes the importance of the high incoherence of amplitude vectors in enhancing the resolution of multi-snapshot spectral estimation. Moreover, to the best of our knowledge, it also provides the first stability result in the super-resolution regime for the well-known sparse MMV problem in DOA estimation.

eess.IV↗
arXiv2025

Universal estimates for the density of states for aperiodic block subwavelength resonator systems

We consider the spectral properties of aperiodic block subwavelength resonator systems in one dimension, with a primary focus on the density of states. We prove that for random block configurations, as the number of blocks $M\to \infty$, the integrated density of states converges to a non-random, continuous function. We show both analytically and numerically that the density of states exhibits a tripartite decomposition: it vanishes identically within bandgaps; it forms smooth, band-like distributions in shared pass bands (a consequence of constructive eigenmode interactions); and, most notably, it exhibits a distinct fractal-like character in hybridisation regions. We demonstrate that this fractal-like behaviour stems from the limited interaction between eigenmodes within these hybridisation regions. Capitalising on this insight, we introduce an efficient meta-atom approach that enables rapid and accurate prediction of the density of states in these hybridisation regions. This approach is shown to extend to systems with quasiperiodic and hyperuniform arrangements of blocks.

math-ph↗
arXiv2025

Nonlinear subwavelength resonances in three dimensions

In this paper, we consider the resonance problem for the cubic nonlinear Helmholtz equation in the subwavelength regime. We derive a discrete model for approximating the subwavelength resonances of finite systems of high-contrast resonators with Kerr-type nonlinearities. Our discrete formulation is valid in both weak and strong nonlinear regimes. Compared to the linear formulation, it characterizes the extra eigenmodes induced by the non-linearlity that have recently been experimentally observed.

math-ph↗
arXiv2025

Space-Time Wave Localisation in Systems of Subwavelength Resonators

In this paper we study the dynamics of metamaterials composed of high-contrast subwavelength resonators and show the existence of localised modes in such a setting. A crucial assumption in this paper is time-modulated material parameters. We prove a so-called capacitance matrix approximation of the wave equation in the form of an ordinary differential equation. These formulas set the ground for the derivation of a first-principles characterisation of localised modes in terms of the generalised capacitance matrix. Furthermore, we provide numerical results supporting our analytical results showing for the first time the phenomenon of space-time localised waves in a perturbed time-modulated metamaterial. Such spatio-temporal localisation is only possible in the presence of subwavelength resonances in the unperturbed structure. We introduce the time-dependent degree of localisation to quantitatively determine the localised modes and provide a variety of numerical experiments to illustrate our formulations and results.

math.AP↗
arXiv2024

Wave packets propagation in the subwavelength regime near the Dirac point

In [Ammari et al., SIAM J Math Anal., 52 (2020), pp. 5441--5466], the first author with collaborators proved the existence of Dirac dispersion cones at subwavelength scales in bubbly honeycomb phononic crystals. In this paper, we study the time-evolution of wave packets that are spectrally concentrated near such conical points. We prove that the wave packets dynamics is governed by a time-dependent effective Dirac system, which still depends, but in a simple way, on the subwavelength scale.

math.AP↗