arXiv subjects

Lingzheng Kong

Publications and source records attributed to Lingzheng Kong.

8 recordsLinked to original sources

arXiv2026

On subwavelength guided seismic waves in Matryoshka-type elastic media with ring defects

Seismic surface waves, particularly low-frequency Rayleigh waves, are notoriously destructive and remain difficult to control using conventional engineering methods. Recent advances in elastic metamaterials have opened new avenues for manipulating and guiding seismic waves at subwavelength scales. In this work, we present a rigorous mathematical study of subwavelength resonances and seismic-wave localization in arbitrarily shaped Matryoshka-type elastic metamaterials, which are composed of nested high-contrast resonators made of materials much stiffer than the background medium. By employing the displacement-to-traction map and a variational formulation, we derive necessary and sufficient conditions that characterize subwavelength resonances. Using the Gohberg--Sigal theory and Puiseux series of multivalued algebraic functions, we establish the existence of subwavelength resonances and obtain their asymptotic characterization in terms of the eigensystem of the generalized stiffness tensor, which serves as the elastic analogue of the capacitance matrix in the celebrated Minnaert acoustic-cavitation systems. Furthermore, in the concentric spherical case, the stiffness tensor exhibits a block-diagonal structure separating translational and rotational components, with each block being tridiagonal and possessing positive, simple eigenvalues. Based on this matrix formulation,we rigorously demonstrate that, for structures with a sufficiently large number of layers, ring defects in layered subwavelength resonators can act as effective radially laminated seismic waveguides, supporting both wave localization and guided propagation along the defects. In particular, appropriately placed ring defects can induce eigenmodes that are exponentially localized at multiple defect sites simultaneously,with precisely quantified amplitude ratios.

math.AP↗

arXiv2025

Inverse conductivity problem with one measurement: Uniqueness of multi-layer structures

In this paper, we study the recovery of multi-layer structures in inverse conductivity problem by using one measurement. First, we define the concept of Generalized Polarization Tensors (GPTs) for multi-layered medium and show some important properties of the proposed GPTs. With the help of GPTs, we present the perturbation formula for general multi-layered medium. Then we derive the perturbed electric potential for multi-layer concentric disks structure in terms of the so-called generalized polarization matrix, whose dimension is the same as the number of the layers. By delicate analysis, we derive an algebraic identity involving the geometric and material configurations of multi-layer concentric disks. This enables us to reconstruct the multi-layer structures by using only one partial-order measurement.

math.AP↗

arXiv2025

Spectral theory of the Neumann-Poincaré operator on multi-layer structures and analysis of plasmon mode splitting

In this paper, we develop a general mathematical framework for analyzing electostatics within multi-layered metamaterial structures. The multi-layered structure can be designed by nesting complementary negative and regular materials together, and it can be easily achieved by truncating bulk metallic material in a specific configuration. Using layer potentials and symmetrization techniques, we establish the perturbation formula in terms of Neumann-Poincaré (NP) operator for general multi-layered medium, and obtain the spectral properties of the NP operator, which demonstrates that the number of plasmon modes increases with the number of layers. Based on Fourier series, we present an exact matrix representation of the NP operator in an apparently unsymmetrical structure, exemplified by multi-layered confocal ellipses. By highly intricate and delicate analysis, we establish a handy algebraic framework for studying the splitting of the plasmon modes within multi-layered structures. Moreover, the asymptotic profiles of the plasmon modes are also obtained. This framework helps reveal the effects of material truncation and rotational symmetry breaking on the splitting of the plasmon modes, thereby inducing desired resonances and enabling the realization of customized applications.

math.AP↗

arXiv2025

Mathematical analysis of subwavelength resonant acoustic scattering in multi-layered high-contrast structures

Multi-layered structures are widely used in the construction of metamaterial devices to realize various cutting-edge waveguide applications. This paper makes several contributions to the mathematical analysis of subwavelength resonances in a structure of $N$-layer nested resonators. Firstly, based on the Dirichlet-to-Neumann approach, we reduce the solution of the acoustic scattering problem to an $N$-dimensional linear system, and derive the optimal asymptotic characterization of subwavelength resonant frequencies in terms of the eigenvalues of an $N\times N$ tridiagonal matrix, which we refer to as the generalized capacitance matrix. Moreover, we provide a modal decomposition formula for the scattered field, as well as a monopole approximation for the far-field pattern of the acoustic wave scattered by the $N$-layer nested resonators. Finally, some numerical results are presented to corroborate the theoretical findings.

math.AP↗

arXiv2024

Mathematical theory on multi-layer high contrast acoustic subwavelength resonators

Subwavelength resonance is a vital acoustic phenomenon in contrasting media. The narrow bandgap width of single-layer resonator has prompted the exploration of multi-layer metamaterials as an effective alternative, which consist of alternating nests of high-contrast materials, called ``resonators'', and a background media. In this paper, we develop a general mathematical framework for studying acoustics within multi-layer high-contrast structures. Firstly, by using layer potential techniques, we establish the representation formula in terms of a matrix type operator with a block tridiagonal form for multi-layer structures within general geometry. Then we prove the existence of subwavelength resonances via Gohberg-Sigal theory, which generalizes the celebrated Minnaert resonances in single-layer structures. Intriguingly, we find that the primary contribution to mode splitting lies in the fact that as the number of nested resonators increases, the degree of the corresponding characteristic polynomial also increases, while the type of resonance (consists solely of monopolar resonances) remains unchanged. Furthermore, we derive original formulas for the subwavelength resonance frequencies of concentric dual-resonator. Numerical results associated with different nested resonators are presented to corroborate the theoretical findings.

math-ph↗

arXiv2023

Elastostatics with multi-layer metamaterial structures and an algebraic framework for polariton resonances

Multi-layer structures are ubiquitous in constructing metamaterial devices to realise various frontier applications including super-resolution imaging and invisibility cloaking. In this paper, we develop a general mathematical framework for studying elastostatics within multi-layer material structures in $\mathbb{R}^d$, $d=2,3$. The multi-layer structure is formed by concentric balls and each layer is filled by either a regular elastic material or an elastic metamaterial. The number of layers can be arbitrary and the material parameters in each layer may be different from one another. In practice, the multi-layer structure can serve as the building block for various material devices. Considering the impingement of an incident field on the multi-layer structure, we first derive the exact perturbed field in terms of an elastic momentum matrix, whose dimension is the same as the number of layers. By highly intricate and delicate analysis, we derive a comprehensive study of the spectral properties of the elastic momentum matrix. This enables us to establishe a handy algebraic framework for studying polariton resonances associated with multi-layer metamaterial structures, which forms the fundamental basis for many metamaterial applications.

physics.class-ph↗

arXiv2022

Positive solutions for a coupled nonlinear Kirchhoff-type system with vanishing potentials

In this paper, we consider the strongly coupled nonlinear Kirchhoff-type system with vanshing potentials: \begin{equation*}\begin{cases} -\left(a_1+b_1\int_{\mathbb{R}^3}|\nabla u|^2\dx\right)Δu+λV(x)u=\fracα{α+β}|u|^{α-2}u|v|^β,&x\in\mathbb{R}^3,\\ -\left(a_2+b_2\int_{\mathbb{R}^3}|\nabla v|^2\dx\right)Δv+λW(x)v=\fracβ{α+β}|u|^α|v|^{β-2}v,&x\in\mathbb{R}^3,\\ u,v\in \mathcal{D}^{1,2}(\R^3), \end{cases}\end{equation*} where $a_i>0$ are constants, $λ,b_i>0$ are parameters for $i=1,2$, $α,β>1$ and $α+β\leqslant 4$, $V(x)$, $W(x)$ are nonnegative continuous potentials, the nonlinear term $F(x,u,v)=|u|^α|v|^β$ is not 4-superlinear at infinity. Such problem cannot be studied directly by standard variational methods, even by restricting the associated energy functional on the Nehari manifold, because Palais-Smale sequences may not be bounded. Combining some new detailed estimates with truncation technique, we obtain the existence of positive vector solutions for the above system when $b_1+b_2$ small and $λ$ large. Moreover, the asymptotic behavior of these vector solutions is also explored as $\textbf{b}=(b_1,b_2)\to \bf{0}$ and $λ\to\infty$. In particular, our results extend some known ones in previous papers that only deals with the case where $4<α+β<6$.

math.AP↗

arXiv2020

Semiclassical solutions for critical Schrödinger-Poisson systems involving multiple competing potentials

In this paper, a class of Schrödinger-Poisson system involving multiple competing potentials and critical Sobolev exponent is considered. Such a problem cannot be studied with the same argument of the nonlinear term with only a positive potential, because the weight potentials set $\{Q_i(x)|1\le i \le m\}$ contains nonpositive, sign-changing, and nonnegative elements. By introducing the ground energy function and subtle analysis, we first prove the existence of ground state solution $v_\varepsilon$ in the semiclassical limit via the Nehari manifold and concentration-compactness principle. Then we show that $v_\varepsilon$ converges to the ground state solution of the associated limiting problem and concentrates at a concrete set characterized by the potentials. At the same time, some properties for the ground state solution are also studied. Moreover, a sufficient condition for the nonexistence of the ground state solution is obtained.

math.AP↗