Journal: Studies in Applied Mathematics

Abbreviation

Stud. appl. math. (Cambr.)

Publisher

Wiley

Journal Volumes

ISSN

0022-2526
1467-9590

Description

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2021 - 20267
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Publications1 - 7 of 7
  • Nonlinear Subwavelength Resonances in Three Dimensions
    Item type: Journal Article
    Ammari, Habib; Kosche, Thea (2025)
    Studies in Applied Mathematics
    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 experimentally observed eigenmodes that are induced by the nonlinearities.
  • High-order exceptional points and enhanced sensing in subwavelength resonator arrays
    Item type: Journal Article
    Ammari, Habib; Davies, Bryn; Orvehed Hiltunen, Erik; et al. (2021)
    Studies in Applied Mathematics
    Systems exhibiting degeneracies known as exceptional points have remarkable properties with powerful applications, particularly in sensor design. These degeneracies are formed when eigenstates coincide, and the remarkable effects are exaggerated by increasing the order of the exceptional point (i.e., the number of coincident eigenstates). In this work, we use asymptotic techniques to study PT-symmetric arrays of many subwavelength resonators and search for high-order asymptotic exceptional points. This analysis reveals the range of different configurations that can give rise to such exceptional points and provides efficient techniques to compute them. We also show how systems exhibiting high-order exceptional points can be used for sensitivity enhancement.
  • Dispersive shock waves for the Boussinesq Benjamin–Ono equation
    Item type: Journal Article
    Nguyen, Lu T.K.; Smyth, Noel F. (2021)
    Studies in Applied Mathematics
    In this work, the dispersive shock wave (DSW) solution of a Boussinesq Benjamin–Ono (BBO) equation, the standard Boussinesq equation with dispersion replaced by nonlocal Benjamin–Ono dispersion, is derived. This DSW solution is derived using two methods, DSW fitting and from a simple wave solution of the Whitham modulation equations for the BBO equation. The first of these yields the two edges of the DSW, while the second yields the complete DSW solution. As the Whitham modulation equations could not be set in Riemann invariant form, the ordinary differential equations governing the simple wave are solved using a hybrid numerical method coupled to the dispersive shock fitting which provides a suitable boundary condition. The full DSW solution is then determined, which is found to be in excellent agreement with numerical solutions of the BBO equation. This hybrid method is a suitable and relatively simple method to fully determine the DSW solution of a nonlinear dispersive wave equation for which the (hyperbolic) Whitham modulation equations are known, but their Riemann invariant form is not.
  • Complex Band Structure for Subwavelength Evanescent Waves
    Item type: Journal Article
    De Bruijn, Yannick; Hiltunen, Erik Orvehed (2025)
    Studies in Applied Mathematics
    We present the mathematical and numerical theory for evanescent waves in subwavelength bandgap materials. We begin in the one-dimensional case, whereby fully explicit formulas for the complex band structure, in terms of the capacitance matrix, are available. As an example, we show that the gap functions can be used to accurately predict the decay rate of the interface mode of a photonic analogue of the Su–Schrieffer–Heeger model. In two dimensions, we derive the bandgap Green's function and characterize the subwavelength gap functions via layer potential techniques. By generalizing existing lattice-summation techniques, we illustrate our results numerically by computing the complex band structure in a variety of settings.
  • Energy Balance and Optical Theorem for Time-Modulated Subwavelength Resonator Arrays
    Item type: Journal Article
    Orvehed Hiltunen, Erik; Rueff, Liora (2026)
    Studies in Applied Mathematics
    We study wave propagation through a one-dimensional array of subwavelength resonators with periodically time-modulated material parameters. Focusing on a high-contrast regime, we use a scattering framework based on Fourier expansions and scattering matrix techniques to capture the interactions between an incident wave and the temporally varying system. This way, we derive a formulation of the total energy flux corresponding to time-dependent systems of resonators. We show that the total energy flux is composed of the transmitted and reflected energy fluxes and derive an optical theorem which characterizes the energy balance of the system. We provide a number of numerical experiments to investigate the impact of the time-dependency, the operating frequency, and the number of resonators on the maximal attainable energy gain and energy loss. Moreover, we show the existence of lasing points, at which the total energy diverges. Our results lay the foundation for the design of energy dissipative or energy amplifying systems.
  • Transmission properties of space-time modulated metamaterials
    Item type: Journal Article
    Ammari, Habib; Cao, Jinghao; Zeng, Xinmeng (2023)
    Studies in Applied Mathematics
    We prove the possibility of achieving exponentially growing wave propagation in space-time modulated media and give an asymptotic analysis of the quasifrequencies in terms of the amplitude of the time modulation at the degenerate points of the folded band structure. Our analysis provides the first proof of existence of k-gaps in the band structures of space-time modulated systems of subwavelength resonators.
  • Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies
    Item type: Journal Article
    Feppon, Florian; Ammari, Habib (2022)
    Studies in Applied Mathematics
    This paper provides several contributions to the mathematical analysis of subwavelength resonances in a high-contrast medium containing N acoustic obstacles. Our approach is based on an exact decomposition formula which reduces the solution of the sound scattering problem to that of an N dimensional linear system, and characterizes resonant frequencies as the solutions to an N-dimensional nonlinear eigenvalue problem. Under a simplicity assumptions on the eigenvalues of the capacitance matrix, we prove the analyticity of the scattering resonances with respect to the square root of the contrast parameter, and we provide a deterministic algorithm allowing to compute all terms of the corresponding Puiseux series. We then establish a nonlinear modal decomposition formula for the scattered field as well as point scatterer approximations for the far-field pattern of the sound wave scattered by N bodies. As a prerequisite to our analysis, a first part of the work establishes various novel results about the capacitance matrix and its symmetry properties, since qualitative properties of the resonances, such as the leading order of the scattering frequencies or of the corresponding far-field pattern are closely related to its spectral decomposition.
Publications1 - 7 of 7