arXiv subjects

Bochao Chen

Publications and source records attributed to Bochao Chen.

13 recordsLinked to original sources

arXiv2026

Symmetrization of the Maxwell--Neumann--Poincar'e operator, spectral decomposition in $\mathbf{H}(\mathrm{curl},D)$ traces, and boundary localisation of SPRs

The Neumann--Poincar\'{e} (NP) operator, a fundamental operator in potential theory, has attracted renewed attention for its central role in the analysis of surface plasmon resonances (SPRs). SPRs, characterized by non-radiative electromagnetic waves at material interfaces with opposing permittivities, underpin advanced technologies such as bio-sensing and cloaking devices. While spectral properties of the scalar NP operator and SPR dynamics for scalar waves are well-established, their vectorial counterparts in Maxwell's framework remain poorly understood. The present work bridges this gap by introducing a symmetrization principle for the matrix-valued Maxwell--Neumann--Poincar\'{e} (MNP) operator, enabling a spectral decomposition of traces in the $\mathbf{H}(\mathrm{curl},D)$ space, which is a foundational advance for electromagnetic theory. Building on this framework, we rigorously characterize the quantum-ergodic localization of the weak plasmon sequence at material boundaries in the full Maxwell system, thereby settling a long-standing question concerning their quantitative description.

math.AP↗

arXiv2025

Quasi-periodic response solutions of nonlinear plate models with nonlocal energy damping

Response solutions are quasi-periodic ones with the same frequency as the forcing term. The present work is devoted to constructing response solutions for $d$-dimensional nonlinear plate models with nonlocal energy damping, which are closely related to damping phenomena in flight structures. For such models, the main characteristic is that the dissipation rate depends on the energy strength. By considering a small parameter $ε$ in the domain excluding the origin and imposing a small quasi-periodic forcing with a Diophantine frequency vector, we demonstrate the persistence of the corresponding response solution. We provide an alternative approach to the contraction mapping principle (cf. [7, 33]) through a combination of reduction together with the Nash--Moser iteration technique. The reason behind this approach lies in the derivative losses caused by the nonlocal nonlinearity.

math.AP↗

arXiv2025

Subwavelength Phononic Bandgaps in High-Contrast Elastic Media

Inspired by [25], this paper investigates subwavelength bandgaps in phononic crystals consisting of periodically arranged hard elastic materials embedded in a soft elastic background medium. Our contributions are threefold. First, we introduce the quasi-periodic Dirichlet-to-Neumann map and an auxiliary sesquilinear form to characterize the subwavelength resonant frequencies, which are identified through the condition that the determinant of a certain matrix vanishes. Second, we derive asymptotic expansions for these resonant frequencies and the corresponding non-trivial solutions, thereby establishing the existence of subwavelength phononic bandgaps in elastic media. Finally, we analyze dilute structures in three dimensions, where the spacing between adjacent resonators is significantly larger than the characteristic size of an individual resonator, allowing the inter-resonator interactions to be neglected. In particular, an illustrative example is presented in which the resonator is modeled as a ball.

math.AP↗

arXiv2025

Analysis of subwavelength resonances in high contrast elastic media by a variational method

In this paper, we present a mathematical study of wave scattering by a hard elastic obstacle embedded in a soft elastic body in three dimensions. Our contributions are threefold. First, we characterize subwavelength resonances using the Dirichlet-to-Neumann map and an auxiliary variational form, showing that these resonances occur when the determinant of a specific matrix vanishes. Second, employing Gohberg-Sigal theory and Puiseux series expansions for multi-valued functions, we derive the asymptotic expansions of subwavelength resonant frequencies in the low-frequency regime through this explicit characterization. Finally, we provide a representation of the scattered field in the interior domain, where the enhancement coefficients are governed by the imaginary parts of the resonant frequencies. Additionally, we establish the transversal and longitudinal far-field patterns for the scattered field in the exterior domain.

math.AP↗

arXiv2023

Resonant modal approximation of time-domain elastic scattering from nano-bubbles in elastic materials

This paper is devoted to establishing the resonant modal expansion of the low-frequency part of the scattered field for acoustic bubbles embedded in elastic materials in the time domain. Due to the nano-bubble with damping, Minnaert resonance can be induced at certain discrete resonant frequencies, which forms the fundamental basis of effectively constructing elastic metamaterials via the composite material theory. There are two major contributions in this work. First, we ansatz a special form of the density, approximate the incident field with a finite number of modes, and then obtain an expansion with a finite number of modes for the acoustic-elastic wave scattering in the time-harmonic regime. Second, we show that the low-frequency part of the scattered field in the time domain can be well approximated by using the resonant modal expansion with sharp error estimates. Interestingly, we find that the $0$-th mode is the main contribution to the resonant modal expansion.

math.AP↗

arXiv2023

Stability for Time-domain Elastic Wave Equations

This paper is concerned with the inverse scattering problem involving the time-domain elastic wave equations in a bounded $d$-dimensional domain. First, an explicit reconstruction formula for the density is established by means of the Dirichlet-to-Neumann operator. The reconstruction is mainly based on the modified boundary control method and complex geometric optics solutions for the elastic wave. Next, the stable observability is obtained by a Carleman estimate. Finally, the stability for the density is presented by the connect operator.

math.AP↗

arXiv2022

Modal approximation for time-domain elastic scattering from metamaterial quasiparticles

This paper aims at quantitatively understanding the elastic wave scattering due to negative metamaterial structures under wide-band signals in the time domain. Specifically, we establish the modal expansion for the time-dependent field scattered by metamaterial quasiparticles in elastodynamics. By Fourier transform, we first analyze the modal expansion in the time-harmonic regime. With the presence of quasiparticles, we validate such an expansion in the static regime via quantitatively analyzing the spectral properties of the Neumann-Poincaré operator associated with the elastostatic system. We then approximate the incident field with a finite number of modes and apply perturbation theory to obtain such an expansion in the perturbative regime. In addition, we give polariton resonances as simple poles for the elastic system with non-zero frequency. Finally, we show that the low-frequency part of the scattered field in the time domain can be well approximated by using the resonant modal expansion with sharp error estimates.

math.AP↗

arXiv2020

Quasi-periodic travelling waves for a class of damped beams on rectangular tori

This article concerns a class of beam equations with damping on rectangular tori. When the generators satisfy certain relationship, by excluding some value of two model parameters, we prove that such models admit small amplitude quasi-periodic travelling wave solutions with two frequencies, which are continuations of two rotating wave solutions with one frequency. This result holds not only for an isotropic torus, but also for an anisotropic torus. The proof is mainly based on a Lyapunov--Schmidt reduction together with the implicit function theorem.

math.DS↗

arXiv2020

Response solutions for wave equations with variable wave speed and periodic forcing

We consider a model of nonlinear wave equations with periodically varying wave speed and periodic external forcing. By imposing non-resonance conditions on the frequency, we establish the existence of the response solutions (i.e., periodic solutions with the same frequency as the forcing) for such a model in a Cantor set of asymptotically full measure. The proof relies on a Lyapunov--Schmidt reduction together with the Nash--Moser iteration.

math.DS↗

arXiv2018

Periodic Solutions to Nonlinear Euler-Bernoulli Beam Equations

Bending vibrations of thin beams and plates may be described by nonlinear Euler-Bernoulli beam equations with $x$-dependent coefficients. In this paper we investigate existence of families of time-periodic solutions to such a model using Lyapunov-Schmidt reduction and a differentiable Nash-Moser iteration scheme. The results hold for all parameters $(ε,ω)$ in a Cantor set with asymptotically full measure as $ε\rightarrow0$.

math.DS↗

arXiv2017

The Existence of Periodic Solutions for Nonlinear Beam Equations on $\mathbb T^d$ by a Para-differential Method

This paper focuses on the construction of periodic solutions of nonlinear beam equations on the $d$-dimensional tori. For a large set of frequencies, we demonstrate that an equivalent form of the nonlinear equations can be obtained by a para-differential conjugation. Given the non-resonant conditions on each finite dimensional subspaces, it is shown that the periodic solutions can be constructed for the block diagonal equation by a classical iteration scheme.

math.DS↗

arXiv2017

Quasi-periodic solutions to nonlinear beam equation on Compact Lie Groups with a multiplicative potential

The goal of this work is to study the existence of quasi-periodic solutions in time to nonlinear beam equations with a multiplicative potential. The nonlinearities are required to only finitely differentiable and the frequency is along a pre-assigned direction. The result holds on any compact Lie group or homogenous manifold with respect to a compact Lie group, which includes the standard torus $\mathbf{T}^{d}$, the special orthogonal group $SO(d)$, the special unitary group $SU(d)$, the spheres $S^d$ and the real and complex Grassmannians. The proof is based on a differentiable Nash-Moser iteration scheme.

math.DS↗

arXiv2017

Periodic Solutions to Nonlinear Wave equation with $X$-dependent Coefficients under the General Boundary Conditions

In this paper we consider a class of nonlinear wave equation with $x$-dependent coefficients and prove existence of families of time-periodic solutions under the general boundary conditions. Such a model arises from the forced vibrations of a nonhomogeneous string and the propagation of seismic waves in nonisotropic media. The proofs are based on a Lyapunov-Schmidt reduction together with a differentiable Nash-Moser iteration scheme.

math.DS↗