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We prove controllability of the Schrödinger equation in ℝd in any time T > 0 with internal control supported on nonempty, periodic, open sets. This demonstrates in particular that controllability of the Schrödinger equation in full space holds for a strictly larger class of control supports than for the wave equation and suggests that the control theory of Schrödinger equation in full space might be closer to the diffusive nature of the heat equation than to the ballistic nature of the wave equation. Our results are based on a combination of Floquet-Bloch theory with Ingham-type estimates on lacunary Fourier series. [ABSTRACT FROM AUTHOR]

This study explores surface-wave propagation through periodic arrangements of multilayered piles in a single-layer soil medium from the viewpoint of bandgaps. The influence of the geometrical parameters of the model on the bandgaps is discussed. The periodic theory of solid-state physics is imposed to derive the dispersion equation. The finite-element method is implemented to calculate the dispersion relation and attenuation zones for surface waves in a periodic pile and soil system. The results show that the bandgaps are sensitive to the size and arrangement of the piles. The practicability of ground vibration-isolation by periodic pile barriers are verified in the time domain. The location and width of bandgaps are further authenticated through finite unit cell–based frequency response analyses. It is found that the frequency range of the Rayleigh wave reduction agrees well with the theoretical attenuation zone. This work provides new insights into the design of periodic piles as wave barriers. This approach has scope for protecting infrastructures, such as nuclear power plants, which pose high environmental risks. It is found that the dispersion analysis gives essential information for Rayleigh wave isolation using piles, including the frequency range and the form of decline. [ABSTRACT FROM AUTHOR]

In this study we consider a wave energy farm consisting of a large number of bottom-hinged paddles. The paddles are arranged in a doubly-periodic manner, with a finite number of identical rows and each row containing an infinite number of paddles. Each paddle is attached to its own damper and spring, allowing power to be generated through its pitching motion. Unlike previous studies into paddle arrays, we envisage compact arrays consisting of small devices arranged across a large number of rows. The mathematical analysis of this proposed configuration is best suited to methods that exploit the periodicity of the array. It is shown that the velocity potential everywhere within the array can be described by an expansion in terms of suitably-defined Floquet–Bloch eigenmodes and this allows the solution to the scattering problem involving waves incident upon the array to be determined by a simple low-order system of equations. The method used in this study is exact within the setting of linearised water waves and has all of the advantages of classical low-frequency multi-scale homogenisation without the low-frequency restriction. It may also be regarded as a natural extension of the well-known wide-spacing approximation without the large separation restriction. Although the focus of this paper is on the mathematical approach to the solution of the problem, we provide a range of results to illustrate the performance of this proposed wave energy converter concept. [ABSTRACT FROM AUTHOR]

This paper presents advanced-kinematics beam models to compute the dispersion characteristics of one-dimensional guides. High-order functions are used to interpolate the primary variables above the waveguide cross-section and along its axis. Taylor- and Lagrange-type bi-dimensional expansions are employed to describe the section deformation, while Lagrangian shape functions approximate the displacement field along the propagating direction. According to the Wave Finite Element Method, the stiffness and mass matrices corresponding to various structural theories are post-processed to build the transfer matrix of a representative waveguide portion. The Carrera Unified Formulation is exploited to calculate these matrices. [ABSTRACT FROM AUTHOR]

This paper develops an asymptotic method to predict the dispersion curves of a weakly disordered medium. The disordered coefficients of the original acoustic equation are related to periodic coefficients through a change of variables, assumed asymptotically close to the identity. At leading order in the amplitude of the disorder, the medium is driven by an acoustic equation with periodic coefficients, which can be analyzed through a classical Floquet–Bloch approach. At first order, with simple eigenvalues, a simple post-processing of the periodic eigenvalues and eigenfunctions of the leading order allows to account for the weakly disordered character of the original coefficients. In the case of repeated eigenvalues (for instance, at Dirac points), a residual is introduced, whose minimization allows to recover the opening of the band gaps with the loss of periodicity. Examples in 1D and 2D illustrate the validity of our asymptotic approach, by comparing its results with eigenmodes and dispersion curves computed over much larger periods for reference. • Dispersion curves and wave modes in weakly non-periodic media are analyzed. • An asymptotic method to predict them is developed. • Opening of the band gaps generated by the loss of periodicity is well reproduced. • Only cell-scale solutions of the periodic medium are needed. [ABSTRACT FROM AUTHOR]

We study the macroscopic transport properties of the quantum Lorentz gas in a crystal with short-range potentials, and show that in the Boltzmann–Grad limit the quantum dynamics converges to a random flight process which is not compatible with the linear Boltzmann equation. Our derivation relies on a hypothesis concerning the statistical distribution of lattice points in thin domains, which is closely related to the Berry–Tabor conjecture in quantum chaos. [ABSTRACT FROM AUTHOR]

A novel method based on waveguide modal analysis is presented to evaluate the effectiveness in terms of frequency bandgaps of bounded periodic metamaterials. The effect of the size of the bounded medium, i.e., the number of periodic unit cells, on the frequency bandgaps predicted by Floquet–Bloch theory in the infinite case is investigated and quantified. The waveguides considered in this work have a bounded cross-section and an infinite longitudinal axis simulated by Perfectly Matched Layers. Its useful area is made of a bounded 1D or 2D-periodic matrix-inclusion metamaterial. In the 2D-periodic case, an approach combining the waveguide modal analysis with the Floquet–Bloch transform is further proposed. Applying the proposed method, the SH-waves propagating in a waveguide are studied by using the finite element method. The discrete spectrum of the waveguide and the associated attenuated or trapped wave modes are calculated and analyzed, which provides, within a semi-analytic framework, an exact characterization regarding the attenuation coefficient and the frequency bandgaps of the studied bounded periodic medium. More particularly, effectiveness indicators are defined to compare the width and the position of frequency bandgaps between the bounded and infinite cases as a function of the cell number. Last but not least, the proposed method is also successfully applied to bounded periodic media with local resonators to characterize their filtering and attenuation effectiveness, which shows its interest in such a case of great practical interest. • A quantitative effectiveness analysis method of bounded metamaterials is proposed. • It is based on waveguide modal analysis and can be coupled with Bloch transform. • Frequency bandgaps for SH-waves in matrix/inclusions periodic media are studied. • Effectiveness indicators are defined to quantify differences with the infinite case. • The minimum cell number to recover the predicted frequency bandgaps is obtained. [ABSTRACT FROM AUTHOR]

• The laser-irradiated kitaev-heisenberg ferromagnet can reveal two topological phases. • The topological phase of the present laser-irradiated ferromagnet can be tuned by the light intensity. • The sign reversal of the thermal hall conductivity is a significant indicator on the photoinduced topological phase transition of the kitaev material. We theoretically study topological properties of Floquet magnons in a laser-irradiated Kitaev-Heisenberg honeycomb ferromagnet with the Dzyaloshinskii-Moriya interaction by making use of the Floquet-Bloch theory. It is found that the Kitaev-Heisenberg honeycomb ferromagnet can reveal two topological phases with different Chern numbers when it is irradiated by the circular-polarized light. We find that the topological phase of the system can be switched from one topological phase to another one by adjusting the light intensity. The intrinsic Dzyaloshinskii-Moriya interaction plays a crucial role in the emergence of photoinduced topological phase transitions. It is shown that sign reversal of the thermal Hall conductivity is an important indicator on photoinduced topological phase transitions in the Kitaev-Heisenberg honeycomb ferromagnet. [ABSTRACT FROM AUTHOR]

Edges of bands of continuous spectrum of periodic structures arise as maxima and minima of the dispersion relation of their Floquet–Bloch transform. It is often assumed that the extrema generating the band edges are non-degenerate. This paper constructs a family of examples of Z 3 -periodic quantum graphs where the non-degeneracy assumption fails: the maximum of the first band is achieved along an algebraic curve of co-dimension 2. The example is robust with respect to perturbations of edge lengths, vertex conditions and edge potentials. The simple idea behind the construction allows generalizations to more complicated graphs and lattice dimensions. The curves along which extrema are achieved have a natural interpretation as moduli spaces of planar polygons. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to construct small-amplitude breather solutions for a nonlinear Klein-Gordon equation posed on a periodic metric graph via spatial dynamics and center manifold reduction. The major difficulty occurs from the irregularity of the solutions. The persistence of the approximately constructed pulse solutions under higher order perturbations is obtained by symmetry and reversibility arguments. [ABSTRACT FROM AUTHOR]

The paper presents a novel comprehensive exact analytical solution for modeling linear shear-horizontal (SH) wave propagation in an isotropic inhomogeneous layer made of functionally graded material, using local Heun functions. The layer is a composite of two materials with varying properties represented by spatial variations following the square of the sine function. The Voigt–Kelvin model is used to account for material losses. The study focuses on SH waves incident at a specific angle and employs the wave splitting technique to analyze forward and backward waves, facilitating the computation of reflection and transmission coefficients at any point in the inhomogeneous structure. The proposed solution utilizes the periodic nature of material functions and employs the Floquet–Bloch theory to derive an exact analytical solution. This approach is particularly suited for cases where SH waves encounter locally periodic functionally graded material. A Riccati equation-based verification is conducted to compare the frequency-dependent modulus of the reflection coefficient obtained from the analytical solution with numerically solved results. The presented work provides a comprehensive and versatile analytical solution for studying linear SH wave propagation in locally inhomogeneous isotropic layers, contributing to the theoretical understanding of elastic wave fields and practical applications. [ABSTRACT FROM AUTHOR]

Low density cellular materials may offer excellent mechanical properties and find wide applicability in lightweight design and infill structures for additive manufacturing, yet currently existing material structures are still far away from their theoretical limit in terms of compressive strength. To explore the existing potential, this paper presents a topology optimization framework for designing periodic cellular materials with maximized strength under compressive loading. Under this condition, the limiting factor of strength is the failure mechanism of buckling instability in the microstructure. In order to predict microstructural buckling, a simplified model based on homogenization theory, a linearized stability criterion and Floquet–Bloch theory is employed. Subsequently, a gradient-based topology optimization problem is formulated to maximize the buckling strength of the most critical failure mode. The framework is utilized to optimize square, triangular and hexagonal microstructures for three different macroscopic load conditions including biaxial, uniaxial and shear loading, and performance assessments are conducted by computation of associated failure surfaces in macroscopic stress space. In all cases, the optimized designs turn out to be first-order hierarchical type microstructures which offer major improvements of strength compared to the initial zero-order designs, however, the gains come at the cost of reductions in stiffness. Furthermore, it is illustrated how imposing geometric symmetry constraints can be exploited to control the shape of the failure surfaces. [ABSTRACT FROM AUTHOR]

We present a procedure for the systematic estimation of the dispersion properties of linear discrete systems with periodic time-varying coefficients. The approach relies on the analysis of a single unit cell, making use of Bloch theorem along with the application of a harmonic balance methodology over an imposed solution ansatz. The solution of the resulting eigenvalue problem is followed by a procedure that selects the eigen-solutions corresponding to the ansatz, which is a plane wave defined by a frequency-wavenumber pair. Examples on spring-mass superlattices demonstrate the effectiveness of the method at predicting the dispersion behavior of linear elastic media. The matrix formulation of the problem suggests the broad applicability of the proposed technique. Furthermore, it is shown how dispersion can inform about the dynamic behavior of time-modulated finite lattices. The technique can be extended to multiple areas of physics, such as acoustic, elastic and electromagnetic systems, where periodic time-varying material properties may be used to obtain non-reciprocal wave propagation. [ABSTRACT FROM AUTHOR]

We study the spectral linear elasticity problem in an unbounded periodic waveguide, which consists of a sequence of identical bounded cells connected by thin ligaments of diameter of order $$ h >0$$ . The essential spectrum of the problem is known to have band-gap structure. We derive asymptotic formulas for the position of the spectral bands and gaps, as $$h \rightarrow 0$$ . [ABSTRACT FROM AUTHOR]

In this paper, we study the spectra of Schrödinger operators on zigzag carbon nanotubes, which are broken by abrasion or during refining process. Throughout this paper, we assume that the carbon nanotubes are broken periodically and we deal with one of those models. Making use of the Floquet-Bloch theory, we examine the spectra of the Schrödinger operators and compare the spectra of the broken case and the pure unbroken case. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the spectrum induced by the supercell method for studying locally defected one-dimensional photonic crystals. By using the propagation matrix method, we show that the continuous spectrum of the periodic structure of the supercell converges to that of the defected photonic crystal. Also, it can be shown that frequencies of localized defected modes in the bandgap are included in a narrow interval (whose length diminishes exponentially with increasing size of the supercell) of the continuous spectrum of the supercell method. [ABSTRACT FROM AUTHOR]

In the limit of an asymptotically large diffusivity ratio of order $\mathcal{O}$(ϵ−2) ≫ 1, steady-state spatially periodic patterns of localized spots, where the spots are centred at lattice points of a Bravais lattice, are well-known to exist for certain two-component reaction–diffusion systems (RD) in $\mathbb{R}$2. For the Schnakenberg RD model, such a localized periodic spot pattern is linearly unstable when the diffusivity ratio exceeds a certain critical threshold. However, since this critical threshold has an infinite-order logarithmic series in powers of the logarithmic gauge ν ≡ −1/log ϵ, a low-order truncation of this series is expected to be in rather poor agreement with the true stability threshold unless ϵ is very small. To overcome this difficulty, a hybrid asymptotic-numerical method is formulated and implemented that has the effect of summing this infinite-order logarithmic expansion for the stability threshold. The numerical implementation of this hybrid method relies critically on obtaining a rapidly converging infinite series representation of the regular part of the Bloch Green's function for the reduced-wave operator. Numerical results from the hybrid method for the stability threshold associated with a periodic spot pattern on a regular hexagonal lattice are compared with the two-term asymptotic results of [10] (Iron et al. J. Nonlinear Science, 2014). As expected, the difference between the two-term and hybrid results is rather large when ϵ is only moderately small. A related hybrid method is devised for accurately approximating the stability threshold associated with a periodic pattern of localized spots for the Gray-Scott RD system in $\mathbb{R}$2. [ABSTRACT FROM PUBLISHER]

The numerical dispersion and dissipation properties of a discontinuous spectral element method are investigated in the context of elastic waves in one dimensional periodic heterogeneous materials. Their frequency dependence and elastic band characteristics are studied. Dispersion relations representing both pass band and stop band structures are derived and used to assess the accuracy of the numerical results. A high-order discontinuous spectral Galerkin method is used to calculate the complex dispersion relations in heterogeneous materials. Floquet-Bloch theory is used to derive the elastic band structure. The accuracy of the dispersion relation is investigated with respect to the spectral polynomial orders for three different cases of materials. Numerical investigations illustrate a spectral convergence in numerical accuracy with respect to the polynomial order based on the elastic band structure and a discontinuous jump of the maximum resolvable frequency within the pass bands resulting in a step-like increase of it with respect to the polynomial order. [ABSTRACT FROM AUTHOR]

We describe explicitly the dispersion relations and spectra of periodic Schrödinger operators on a graphyne nanotube structure. [ABSTRACT FROM AUTHOR]

We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a three-dimensional periodic medium; the defect is infinitely extended in one direction, but compactly supported in the remaining two. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We will show that even small perturbations lead to additional spectrum in the spectral gaps of the unperturbed operator and investigate some properties of the spectrum that is created. [ABSTRACT FROM AUTHOR]

The linear stability of steady-state periodic patterns of localized spots in $${\mathbb {R}}^2$$ for the two-component Gierer-Meinhardt (GM) and Schnakenberg reaction-diffusion models is analyzed in the semi-strong interaction limit corresponding to an asymptotically small diffusion coefficient $${\displaystyle \varepsilon }^2$$ of the activator concentration. In the limit $${\displaystyle \varepsilon }\rightarrow 0$$ , localized spots in the activator are centered at the lattice points of a Bravais lattice with constant area $$|\Omega |$$ . To leading order in $$\nu ={-1/\log {\displaystyle \varepsilon }}$$ , the linearization of the steady-state periodic spot pattern has a zero eigenvalue when the inhibitor diffusivity satisfies $$D={D_0/\nu }$$ for some $$D_0$$ independent of the lattice and the Bloch wavevector $${\pmb k}$$ . From a combination of the method of matched asymptotic expansions, Floquet-Bloch theory, and the rigorous study of certain nonlocal eigenvalue problems, an explicit analytical formula for the continuous band of spectrum that lies within an $${\mathcal O}(\nu )$$ neighborhood of the origin in the spectral plane is derived when $$D={D_0/\nu } + D_1$$ , where $$D_1={\mathcal O}(1)$$ is a detuning parameter. The periodic pattern is linearly stable when $$D_1$$ is chosen small enough so that this continuous band is in the stable left half-plane $$\text{ Re }(\lambda )<0$$ for all $${\pmb k}$$ . Moreover, for both the Schnakenberg and GM models, our analysis identifies a model-dependent objective function, involving the regular part of the Bloch Green's function, that must be maximized in order to determine the specific periodic arrangement of localized spots that constitutes a linearly stable steady-state pattern for the largest value of $$D$$ . From a numerical computation, based on an Ewald-type algorithm, of the regular part of the Bloch Green's function that defines the objective function, it is shown within the class of oblique Bravais lattices that a regular hexagonal lattice arrangement of spots is optimal for maximizing the stability threshold in $$D$$ . [ABSTRACT FROM AUTHOR]

We study a class of periodic Schrödinger operators on R that have Dirac points. The introduction of an "edge" via adiabatic modulation of a periodic potential by a domain wall results in the bifurcation of spatially localized "edge states," associated with the topologically protected zero-energy mode of an asymptotic one-dimensional Dirac operator. The bound states we construct can be realized as highly robust transverse-magnetic electromagnetic modes for a class of photonic waveguides with a phase defect. Our model captures many aspects of the phenomenon of topologically protected edge states for 2D bulk structures such as the honeycomb structure of graphene. [ABSTRACT FROM AUTHOR]

In this paper, a Floquet Model Order Reduction (MOR) method is proposed for the modelling of finite periodic structures subjected to harmonic or transient loads. This Floquet MOR generates frequency-independent projection matrices of the full-scale structure using a Floquet expansion of the local state-vectors computed on a single unit-cell with the Forced Wave Finite Element framework. This two-step MOR method can handle large multi-scale waveguides regardless of the scale and the number of unit-cells. A comparison with a standard sub-structuring approach based on unit-cell wave-mode reduction demonstrates that Floquet MOR is able to create reduced models of finite periodic structures with an unequalled computational efficiency. The method's ability to produce small and accurate models suitable for time-domain simulations presents a high potential for virtual sensing, real-time Structural Health Monitoring (SHM) and digital twins for large periodic structures and metamaterials. • We introduce an original MOR technique for transient simulations in periodic media. • The reduced model of the overall structure is built from a single unit-cell analysis. • The numerical framework is based on WFEM and exploits a POD-like Floquet expansion. • The MOR handles complex waveguiding effects and is validated on a metamaterial model. [ABSTRACT FROM AUTHOR]

The Floquet-Bloch theory (FBT) is used to determine the radiation loss of a grating-assisted directional coupler. Improper waves obtained by FBT represent leakage and produce radiation loss along the transverse direction. By summing the fields of all leaky harmonics, the power flow radiated into the superstrate and substrate can be calculated from Poynting's theorem. However, the locations at which these summations occur impact the accuracy. An analytic formula for radiation loss is also derived. Results from both the FBT and the analytic formula are compared with results obtained using the coupled-mode theory. [ABSTRACT FROM PUBLISHER]

In this paper, a two-dimensional (2-D) method describing the whispering gallery mode in a microgear resonator is presented. The microgear is a microdisk surrounded by a circular grating. The method, which is based on the Floquet-Bloch formalism, analytically describes the field within the disk and outside the grating. On the other hand, the field within the grating is calculated using a finite-difference scheme in polar coordinates. Matching the boundary conditions, it is possible to work in a forced oscillation regime or in a free oscillation regime (laser mode). The resonant wavelength and quality factor can then be deduced. Compared to the coupled mode theory and to 2-D finite-difference time-domain computations, the method is faster and more accurate. Moreover, a polarization effect of the microgear is demonstrated. The It polarization experiences a Q-factor improvement contrary to TM polarization. Finally, microgear structures prove to be more efficient than micro flowers. [ABSTRACT FROM AUTHOR]