arXiv subjects

Mourad Sini

Publications and source records attributed to Mourad Sini.

At least 19 recordsLinked to original sources

arXiv2026

Eigenvalue Asymptotics in High-Contrast Media

We investigate the eigenfrequency asymptotics of a bounded acoustic cavity containing a shrinking high-contrast inclusion. This configuration is motivated by contrast-enhanced ultrasound imaging, in which microbubbles are employed as acoustic contrast agents. We identify dimension-dependent material scalings that keep the inclusion-induced resonance in a fixed order-one frequency regime as the inclusion shrinks. Our main result gives a complete asymptotic description of the spectrum in any prescribed bounded frequency window. Two distinct spectral mechanisms arise: the background Dirichlet eigenfrequencies persist as perturbed eigenvalue clusters, while the singular material contrast creates an additional Minnaert eigenvalue branch. The limiting Minnaert frequency is explicit in both dimensions, with a capacitance-based expression in three dimensions and an area-based expression in two dimensions. Two-dimensional numerical experiments confirm both the perturbation of the background Dirichlet eigenfrequencies and the emergence of the additional Minnaert branch.

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arXiv2026

A Time-Domain Pressure-Interface Model for Gas Bubble Dynamics with Surface Tension: Well-Posedness, Classical Limits, and Resonance Branches

We derive and analyze a time-domain pressure--interface evolution model for a compressible gas bubble in a compressible liquid with surface tension. Starting from the nonlinear two-phase Euler free-boundary problem and linearizing about a spherical Young--Laplace equilibrium, we obtain a coupled bulk--surface hyperbolic system for the liquid pressure, the gas pressure, and the normal displacement of the interface. The time-domain analysis is complicated by the fact that the surface-tension quadratic form is indefinite: the $Y_0^0$ component is the volume-changing breathing mode, the $Y_1^m (m = \{-1,0,1\})$ components are neutral translations, and only the higher spherical harmonics give a coercive shape-mode energy. We exploit this decomposition. The coercive sector is treated by a $C^0$-semigroup argument, while the breathing mode is analyzed separately by Fourier--Laplace methods. This yields well-posedness for admissible finite-energy data and classical solutions under the natural compatibility conditions. We then justify two limiting descriptions with quantitative error estimates. In an acoustic quasi-static regime, the model reduces to the linearized Rayleigh--Plesset equation for the breathing mode and to the linearized Rayleigh--Lamb equations for the shape modes. In a different regime, it reduces to the frozen-interface acoustic transmission model. Finally, a frequency-domain analysis identifies the corresponding Minnaert, Rayleigh--Lamb, and Fabry--P\'erot-type resonance mechanisms as different components and limits of the same pressure--interface formulation.

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arXiv2026

Finite Spectral-Band Optimal Control of Acoustic Waves via Subwavelength Point-Like Resonant Actuators

We study finite-band optimal control of acoustic waves actuated by local clusters of subwavelength resonators. The acoustic problem reduces to a time-domain Foldy-Lax approximation capturing wave-structure interaction. Spectral analysis of the delayed transfer matrix isolates collective scattering resonances corresponding to weakly damped poles $s_\alpha^\epsilon=-\gamma_\alpha^\epsilon+i\omega_\alpha^\epsilon$ with radiation damping $\gamma_\alpha^\epsilon>0$. Projecting onto a finite band yields the coupled system $\ddot{a}+\Lambda a=C_\epsilon\eta$, $\ddot{\eta}+2\Gamma_\epsilon\dot{\eta}+K_\epsilon\eta=u$, where $a$, $\eta$, and $u$ are modal coefficients, microstructural states, and control. For a tracking functional $\mathcal{J}_{\mu}$ with regularization $\mu>0$, we prove existence and uniqueness of the optimal control and derive the adjoint system. Our main quantitative result is a resonant source-lifting estimate: if a source profile $\eta_r$ is spectrally concentrated in bands $I_\alpha$, the input $u_r=(\partial_t^2+2\Gamma_\epsilon\partial_t+K_\epsilon)\eta_r$ satisfies $\|u_r\|_{L^2(0,T)}^2 \le \sum_\alpha \left(\sup_{\nu\in I_\alpha} |(\omega_\alpha^\epsilon)^2+(\gamma_\alpha^\epsilon)^2-\nu^2 +2i\gamma_\alpha^\epsilon\nu|\right)^2 \|(\eta_r)_\alpha\|_{\mathcal{B}_T(I_\alpha)}^2$. This provides an upper bound for the optimal value function. At exact matching $\nu=\omega_\alpha^\epsilon$, the multiplier equals $2\gamma_\alpha^\epsilon\omega_\alpha^\epsilon+O((\gamma_\alpha^\epsilon)^3)$, showing clustering yields a finite resonant gain governed by the pole's real part. Finally, this attenuation enables finite-band stabilization under an explicit modal coupling condition, with a decay rate proportional to the cluster damping scale.

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arXiv2026

Resonant Microstructures as Dirac-type Actuators for Acoustic Wave Control

We study interior control of the acoustic wave equation via effective point sources generated by a finite cluster of resonant perturbations (modeling acoustic subwavelength bubbles). At the abstract level, after localizing the whole-space dynamics to a large auxiliary observation domain, we consider a Dirichlet spectral formulation of the wave equation with finitely many point actuators located at prescribed interior points. Restricting to a finite spectral band of Dirichlet eigenfrequencies, we prove that, under a natural full-rank condition on the associated coupling matrix, arbitrary trajectories on the corresponding spectral subspace can be realized, with quantitative bounds on the control cost in terms of spectral-band geometry and actuator placement. We then show that these ideal actuators can be realized by clusters of small, high-contrast bubbles. Using a time-domain asymptotic expansion, the scattered field is represented as a superposition of retarded monopoles whose amplitudes satisfy a finite-dimensional delayed hyperbolic system. In the Laplace domain, this induces a transfer operator whose pole structure encodes the Minnaert resonance with a collective attenuation. We prove that the associated actuator map is ill-conditioned away from resonance, whereas, under a cluster-level transducer accessibility condition linking the incident fields to the dominant cluster channels, it admits a bounded right inverse on suitable Minnaert bands. Consequently, one obtains spectral tracking of the wave field with error $\mathcal{O}(\varepsilon^\gamma)$ as the bubble size $\varepsilon \to 0$. Keywords. Wave equation, Dirac actuators, Trajectory tracking control, Resonant perturbations, Kato's analytic perturbation, Perron-Frobenius spectrum, Minnaert resonances, Actuation map, Toeplitz matrix.

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arXiv2026

Feedback Stabilization and Tracking for Heat Equations Using Thermo-Plasmonic Nanoparticles as Actuators

We propose a feedback strategy to track prescribed heat profiles using plasmonic nanoparticles as actuators. Starting from a thermo--plasmonic Maxwell--heat model, we use a time-domain discrete effective description in which the generated heat is approximated by a superposition of heat kernels centered at particle locations with amplitudes governed by a coupled Volterra system. We recast this dynamics as a heat equation on a bounded domain with finitely many point actuators and design a tracking feedback based on pointwise evaluations of $\mathcal A^{-1}y$, where $\mathcal A=I-A_0$ and $A_0$ is the Neumann diffusion operator. Working in the natural $V'$ setting with $V=D(\mathcal A)$, we prove exponential stabilization of the tracking error via distribution-actuator theory. For non-equilibrium reference profiles, we add a constant feedforward term and a low-mode fixed-point pre-compensation on $X_N$, ensuring exact steady matching on $X_N$ and an explicit bound on the residual tail mismatch.

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arXiv2026

Efficient Numerical Reconstruction of Wave Equation Sources via Droplet-Induced Asymptotics

In this paper, we develop and numerically implement a novel approach for solving the inverse source problem of the acoustic wave equation in three dimensions. By injecting a small high-contrast droplet into the medium, we exploit the resulting wave field perturbation measured at a single external point over time. The method enables stable source reconstructions where conventional approaches fail due to ill-posedness, with potential applications in medical imaging and non-destructive testing. Key contributions include: 1. Implementation of a theoretically justified asymptotic expansion, from [33], using the eigensystem of the Newtonian operator, with error analysis for the spectral truncation. 2. Novel numerical schemes for solving the time-domain Lippmann-Schwinger equation and reconstructing the source via Riesz basis expansions and mollification-based numerical differentiations. 3. Reconstruction requiring only single-point measurements, overcoming traditional spatial data limitations. 4. 3D numerical experiments demonstrating accurate source recovery under noise (SNR of the order $1/a$), with error analysis for the droplet size (of the order $a$) and the number of spectral modes $N$.

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arXiv2026

Elastic Calder\'on Problem via Resonant Hard Inclusions: Linearisation of the N-D Map and Density Reconstruction

We study an elastic Calderon-type inverse problem: recover the mass density $\rho(x)$ in a bounded domain $\Omega\subset\mathbb{R}^3$ from the Neumann-to-Dirichlet map associated with the isotropic Lam\'e system $\mathcal{L}_{\lambda,\mu}u+\omega^2\rho(x)u=0$. We introduce a constructive strategy that embeds a subwavelength periodic array of resonant high-density (hard) inclusions to create an effective medium with a uniform negative density shift. Specifically, we place a periodic cluster of inclusions of size $a$ and density $\rho_1\asymp a^{-2}$ strictly inside $\Omega$. For frequencies $\omega$ tuned to an eigenvalue of the elastic Newton (Kelvin) operator of a single inclusion, we show that as $a\to0$ and the number of inclusions $M\to\infty$, the Neumann-to-Dirichlet map $\Lambda_D$ converges to an effective map $\Lambda_{\mathcal{P}}$ corresponding to a background density shift $-\mathcal{P}^2$, with the operator norm estimate $\|\Lambda_D-\Lambda_{\mathcal{P}}\|\le Ca^{\alpha}\mathcal{P}^6$ for some $\alpha>0$ determined by the geometric scaling. Around this negative background we derive a first-order linearization of $\Lambda_{\mathcal{P}}$ in terms of $\rho$ and the Newton volume potential for the shifted Lam\'e operator. Testing the linearized relation with complex geometric optics solutions yields an explicit reconstruction formula for the Fourier transform of $\rho$, and hence a global density recovery scheme. The results provide a metamaterial-inspired analytic framework for inverse coefficient problems in linear elasticity and a concrete paradigm for leveraging nanoscale resonators in reconstruction algorithms.

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arXiv2026

High-Contrast Transmission Resonances for the Lam\'e System

We consider the Lam\'e transmission problem in $\mathbb{R}^3$ with a bounded isotropic elastic inclusion in a high-contrast setting, where the interior-to-exterior Lam\'e moduli and densities scale like $1/\tau$ as $\tau\to0$. We study the scattering resonances of the associated self-adjoint Hamiltonian, defined as the poles of the meromorphic continuation of its resolvent. We obtain a sharp asymptotic description of resonances near the real axis as $\tau\to0$. Near each nonzero Neumann eigenvalue of the interior Lam\'e operator there is a cluster of resonances lying just below it in the complex plane; in this wavelength-scale regime the imaginary parts are of order $\tau$ with non-vanishing leading coefficients. In addition, near zero (a subwavelength regime), we identify resonances with real parts of order $\sqrt{\tau}$ and prove a lifetime dichotomy: their imaginary parts are of order $\tau$ generically, but of order $\tau^2$ for an explicit admissible set $\mathcal E$. This yields a classification of long-lived elastic resonances in the high-contrast limit. We also establish resolvent asymptotics for both fixed-size resonators and microresonators. We derive explicit expansions with a finite-rank leading term and quantitative remainder bounds, valid near both wavelength-scale and subwavelength resonances. For microresonators, at the wavelength scale the dominant contribution is an anisotropic elastic point scatterer. Near the zero eigenvalue, the leading-order behaviour is of monopole or dipole type, and we give a rigorous criterion distinguishing the two cases.

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arXiv2026

Electromagnetic Scattering by a Cluster of Hybrid Dielectric-Plasmonic Dimers

We consider time-harmonic electromagnetic scattering by a cluster of hybrid dielectric-plasmonic dimers in $\mathbb{R}^3$. Each dimer consists of a high-contrast dielectric nanoparticle and a moderately contrasting plasmonic nanoparticle separated by a subwavelength distance. The cluster is assumed to contain many such dimers whose size $a$ is small compared to the wavelength, with intra-dimer and inter-dimer distances scaling like $a^{t_1}$ and $a^{t_2}$, and the frequency is tuned near suitable electric and magnetic resonances of the associated Newtonian and magnetization operators on the reference shapes. Under these geometric, contrast and spectral assumptions, we derive a Foldy--Lax type approximation for the Maxwell system. We show that the scattered field and its far field admit asymptotic expansions in terms of four moments attached to each dimer, which solve an explicit finite-dimensional linear system. We prove invertibility of this system under quantitative smallness conditions on the contrast and the dimer density, and we obtain error estimates uniform in the number of dimers. By extracting the dominant components, we further show that each hybrid dimer behaves, at leading order, as a co-located electric and magnetic dipole driven by the local fields, and we identify the corresponding $6\times 6$ polarizability matrix. This provides a discrete model for clusters of hybrid dimers that is suitable for fast forward simulations, inverse schemes, and as input for effective-medium descriptions. In particular, it suggests parameter regimes where clusters of hybrid dimers can generate (double) negative effective permittivity and permeability and bi-anisotropic constitutive laws and eventually hyperbolic media.

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arXiv2025

Functional-Analytic Justification of the Time-Domain Foldy-Lax Approximation for Dispersive Acoustic Media: A Feynman-Diagram Viewpoint

This work provides a rigorous functional-analytic justification for a time-domain Foldy-Lax framework that describes multiple acoustic scattering by a cluster of dispersive resonators (modeling gas-filled bubbles), explicitly incorporating dispersion via the Minnaert resonance. The model is formulated as a delayed-coupled hyperbolic system for bubble amplitude interactions. We combine time-domain integral equations, Laplace transforms, and Hardy-Sobolev space techniques to analyze this system, establishing its unique solvability in anisotropic Hilbert spaces, with solutions expressed as convergent Neumann series of convolution operators. We derive geometric decay of truncation errors for resonant incident waves and quantify the contribution of $N$-th order multi-scattering, showing it scales with $\varepsilon^{N(1-p)+1}$ (relating bubble radius $\varepsilon$ and inter-bubble distance scaling as $\varepsilon^p$, $p<1$). This dominates the measurement errors, which are of order $\varepsilon^2$, thereby allowing us to capture fields generated by inter-bubble interactions of order $N<\frac{1}{1-p}$. This provides a quantitative relation between the spectra band width of the source field, the closeness distance between the bubbles and the order $N$ of the relevant interactions between the bubbles. Furthermore, a novel connection to Feynman diagrams maps multi-scattering paths to diagrammatic vertices and propagators, simplifying the interpretation of higher-order interactions and kinematic constraints. This framework advances accurate transient wave prediction in dispersive media, with implications for cavitation therapy, seismic imaging, and metamaterial engineering.

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arXiv2025

High Contrast Transmission and Fabry-P\'erot-type Resonances

It is well known, in the acoustic model, that highly contrasting transmission leads to the so-called Minnaert subwavelength resonance. In this work, we show that such highly contrasting transmissions create not only one resonance but a family of infinite resonances located near the real axis where the first one (i.e. the smallest) is indeed the Minnaert one. This family of resonances are the shifts (in the lower complex plan) of the Neumann eigenvalues of the Laplacian. The well known Minneart resonance is nothing but the shift of the trivial (zero) Neumann eigenvalue of the bubble. These resonances, other than the Minnaert ones, are Fabry-P\'erot-type resonances as the generated total fields, in the bubble, are dominated by a linear combination of the Neumann eigenfunctions which, in particular, might create interferences. In addition, we establish the following properties. 1. We derive the asymptotic expansions, at the second order, of this family of resonances in terms of the contrasting coefficient. 2. In the time-harmonic regime, we derive the resolvent estimates of the related Hamiltonian and the asymptotics of scattered fields that are uniform in the whole space, highlighting the contributions from this sequence of resonances. 3. In the time domain regime, we derive the time behavior of the acoustic microresonator at large time-scales inversely proportional to powers of microresonator's radius. 4. The analysis shows that near Fabry-P\'erot resonances, the mircoresonator exhibits pronounced anisotropy. We believe that such a feature may pave the way for designing anisotropic metamaterials from simple configurations of a single microresonator.

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arXiv2025

The Calderon Problem Revisited: Reconstruction With Resonant Perturbations

The original Calderón problem consists in recovering the potential (or the conductivity) from the knowledge of the related Neumann to Dirichlet map (or Dirichlet to Neumann map). Here, we first perturb the medium by injecting small-scaled and highly heterogeneous particles. Such particles can be bubbles or droplets in acoustics or nanoparticles in electromagnetism. They are distributed, periodically for instance, in the whole domain where we want to do reconstruction. Under critical scales between the size and contrast, these particles resonate at specific frequencies that can be well computed. Using incident frequencies that are close to such resonances, we show that 1) the corresponding Neumann to Dirichlet map of the composite converges to the one of the homogenized medium. In addition, the equivalent coefficient, which consist in the sum of the original potential and the effective coefficient, is negative valued with a controlable amplitude. 2) as the equivalent coefficient is negative valued, then we can linearize the corresponding Neumann to Dirichlet map using the effective coefficient's amplitude. 3) from the linearized Neumann to Dirichlet map, we reconstruct the original potential using explicit complex geometrical optics solutions (CGOs).

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arXiv2024

Dispersive Effective Metasurface Model for Bubbly Media

We derive the effective transmission condition for a cluster of acoustic subwavelength resonators, modeled as small-scaled bubbles distributed not necessarily periodically along a smooth, bounded hypersurface, which need not be flat. The transmission condition specifies that the jump in the normal derivative of the acoustic field is proportional to its second time derivative, convoluted in time with a sinusoidal kernel. This kernel has a period determined by the common subwavelength resonance (specifically, the Minnaert resonance in this case). This dispersive transmission condition can also be interpreted as a Dirac-like surface potential that is convoluted in the time domain and spatially supported on the specified hypersurface. We highlight the following features: 1. High resonance regime: When the common resonance is large, the surface behaves as fully transparent, permitting complete transmission of the acoustic field. 2. Moderate resonance regime: For moderate resonance values, the surface acts as a screen with memory effects, capturing the dispersive behavior induced by the resonance. 3. Low resonance regime: When the common resonance is small, the surface functions as a partial reflective (or partial transmissive) screen with no memory effect.

math.AP↗

arXiv2024

Electromagnetic waves generated by a hybrid dielectric-plasmonic dimer

We know that the electric field generated by a plasmonic nano-particle (with negative permittivity) is given as a polarization of the incident electric field. Similarly, the electric field produced by a dielectric nano-particle (with positive but high permittivity) is given as a polarization of the incident magnetic field. In this work, we demonstrate that a hybrid dimer composed of two closely coupled nano-particles, one plasmonic and the other dielectric can polarize both the incident electric and magnetic fields. Consequently, such hybrid dimers have the potential to modify both the electric permittivity and magnetic permeability of the surrounding medium. However, this dual modification occurs only when the two nano-particles share common resonant frequencies. We derive the asymptotic expansion of the fields generated by these hybrid dimers in the subwavelength regime for incident frequencies near their shared resonant frequencies.

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arXiv2024

Effective Medium Theory for Heat Generation Using Plasmonics: A Parabolic Transmission Problem Driven by the Maxwell System

The excitation of plasmonic nanoparticles by incident electromagnetic waves at frequencies near their subwavelength resonances induces localized heat generation in the surrounding medium. We develop a mathematical framework to rigorously quantify this heat generation in systems of arbitrarily distributed nanoparticles. 1. For an arbitrary discrete distribution of M nanoparticles within a bounded domain, the effective heat distribution is described by a coupled system: Volterra-type integral equations for the heat conduction and a Foldy-Lax-type system governing the self consistent electric field intensities. These equations are parameterized by the particle geometries and the local electromagnetic field interactions. The effective heat generation is computed by solving these coupled systems, with the computational complexity scaling as M^2. 2. In the case M >> 1, under natural scaling regimes, the discrete system converges to a continuum model, yielding an effective parabolic equation for the heat distribution. The source term in this homogenized parabolic model is characterized by the solution of the homogenized Maxwells equations, incorporating an effective permittivity distribution derived from the Drude model under resonance conditions. Our analysis utilizes advanced tools in potential theory, asymptotic analysis and homogenization. By leveraging layer potential representations, we derive point-wise field approximations. The coupling between the Maxwell and heat equations is resolved by analyzing the spectral properties of the nanoparticles and their scaling limits. This framework reduces the problem to two mathematical challenges: a control problem for the effective parabolic system and an internal phase-less inverse problem for the Maxwell system, thus providing a unified approach to modeling heat generation in nanoparticle clusters.

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arXiv2024

Uniform Space and Time Behavior for Acoustic Resonators

We deal with the time-domain acoustic wave propagation in the presence of a subwavelength resonator given by a Minneart bubble. This bubble is small scaled and enjoys high contrasting mass density and bulk modulus. It is well known that, under certain regimes between these scales, such a bubble generates a single low-frequency (or subwavelength) resonance called Minnaert resonance. In this paper, we study the wave propagation governed by Minnaert resonance effects in time domain. We derive the point-approximation expansion of the wave field. The dominant part is a sum of two terms. 1. The first one, which we call the primary wave, is the wave field generated in the absence of the bubble. 2. The second one, which we call the resonant wave, is generated by the interaction between the bubble and the background. It is related to a Dirac-source, in space, that is modulated, in time, with a coefficient which is a solution of a $1$D Cauchy problem, for a second order differential equation, having as propagation and attenuation parameters the real and the imaginary parts, respectively, of the Minnaert resonance. We show that the evolution of the resonant wave remains valid for a large time of the order $\epsilon^{-1}$, where $\epsilon$ is the radius of the bubble, after which it collapses by exponentially decaying. Precisely, we confirm that such resonant wave have life-time inversely proportional to the imaginary part of the related subwavelength resonances, which is in our case given by the Minnaert one. In addition, the real part of this resonance fixes the period of the wave.

math.AP↗

arXiv2024

Time-Dependent Acoustic Waves Generated by Multiple Resonant Bubbles: Application to Acoustic Cavitation

We analyse the ultrasound waves reflected by multiple bubbles in the linearized time-dependent acoustic model. The generated time-dependent wave field is estimated close to the bubbles. The motivation of this study comes from the therapy modality using acoustic cavitation generated by injected bubbles into the region of interest. The goal is to create enough, but not too much, pressure in the region of interest to eradicate anomalies in that region. Here, we derive the dominant part of the generated acoustic field by a cluster of bubbles taking into account the (high) contrasts of their mass density and bulk as well as their general distribution in the given region. As consequences of these approximations, we highlight the following features: 1. If we use dimers (two close bubbles), or generally polymers, then we obtain a remarkable enhancement of the whole echo in the whole time. 2. If we distribute the bubbles every where in the region of interest,then we can derive the effective acoustic model which turns out to be a dispersive one. We show that, for a given desired pressure, we can tune the effective model to generate it.

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arXiv2024

Extraction of the mass density using only the ${\mathtt{p}}$-parts of the elastic fields generated by injected highly dense small inclusions

We propose a reconstruction method to extract the variable mass density from the elastic farfields, with a single incident direction, measured before and after injecting highly dense small scaled inclusions. We take as a model, the Lamé system where the mass density is the unknown in $Ω$ and the Lamé parameters are known constants. The injected small/dense inclusion, $D:=z +a B\, (\subset\subset Ω)$ with $z$ as its location, $a\ll 1$ as its maximum radius and $B$ of unit volume, generates a sequences of resonant frequencies. These special frequencies are related to the eigenvalues of the Lamé volume integral operator defined on the domain of the inclusion and thus are, in principle, computable. After injecting the small inclusion at a location point $z$, we send an elastic incident plane wave at an incident frequency close to one of the mentioned resonant frequencies, say $ω_{n_0}$. Contrasting the ($\mathtt{p}$-parts of the) farfields generated, at one incident direction, before and after injecting this small inclusion, we provide an explicit formula that allows us to recover the total field $V^{t,\mathtt{p}}(z,-\hat{x})$ corresponding to $\mathtt{p}$-incident waves at the location $z$. This total field is generated in the absence of the inclusion. Then we repeat the experiment by injecting more inclusions inside $Ω$. Using this reconstructed field in the Lamé PDE system, via a numerical differentiation, we recover the values of the mass density inside $Ω$. It is worth mentioning that, we use measurements of dimension 3 to recover a function of 3 dimensions freedom. This makes the inverse problem not over determined. In addition, we use only the pressure wave and the $\mathtt{p}$-part of the farfield, for the reconstruction. To our best knowledge, this is the first result using only one type of elastic waves for the parameter identification.

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