Mathieu Functions Research Articles

Single-Qubit Driving Fields and Mathieu Functions

We report a new family of time-dependent single-qubit radiation fields for which the correspondent evolution operator can be disentangled in an exact way via the Wei–Norman formalism. Such fields are characterized in terms of the Mathieu functions. We show that the regions of stability of the Mathieu functions determine the nature of the driving fields: For parameters in the stable region, the fields are oscillating, being able to be periodic under certain conditions. Whereas, for parameters in the instability region, the fields are pulse-like. In addition, in the stability region, this family admits solutions for evolution loops in quantum control. We obtain some prescriptions to reach such a control effect. Geometric phases in the evolution are also analyzed and discussed.

Dynamical band gap tuning in anisotropic tilted Dirac semimetals by intense elliptically polarized normal illumination and its application to8−Pmmnborophene

The Dynamical-gap formation in Weyl semimetals modulated by intense elliptically polarized light is addressed through the solution of the time-dependent Schr\"odinger equation for the Weyl Hamiltonian via the Floquet theorem. The time-dependent wave functions and the quasi-energy spectrum of the two-dimensional Weyl Hamiltonian under normal incidence of elliptically polarized electromagnetic waves are obtained using a non-perturbative approach. In it, the Weyl equation is reduced to an ordinary second-order differential Mathieu equation. It is shown that the stability conditions of the Mathieu functions are directly inherited by the wave function resulting in a quasiparticle spectrum consisting of bands and gaps determined by dynamical diffraction and resonance conditions between the electron and the electromagnetic wave. Estimations of the electromagnetic field intensity and frequency, as well as the magnitude of the generated gap are obtained for the $8-Pmmn$ phase of borophene. We provide with a simple method that enables to predict the formation of dynamical-gaps of unstable wave functions and their magnitudes. This method can readily be adapted to other Weyl semimetals.

Balancing long-range interactions and quantum pressure: Solitons in the Hamiltonian mean-field model.

The Hamiltonian mean-field (HMF) model describes particles on a ring interacting via a cosine interaction, or equivalently, rotors coupled by infinite-range XY interactions. Conceived as a generic statistical mechanical model for long-range interactions such as gravity (of which the cosine is the first Fourier component), it has recently been used to account for self-organization in experiments on cold atoms with long-range optically mediated interactions. The significance of the HMF model lies in its ability to capture the universal effects of long-range interactions and yet be exactly solvable in the canonical ensemble. In this work we consider the quantum version of the HMF model in one dimension and provide a classification of all possible stationary solutions of its generalized Gross-Pitaevskii equation (GGPE), which is both nonlinear and nonlocal. The exact solutions are Mathieu functions that obey a nonlinear relation between the wave function and the depth of the mean-field potential, and we identify them as bright solitons. Using a Galilean transformation these solutions can be boosted to finite velocity and are increasingly localized as the mean-field potential becomes deeper. In contrast to the usual local GPE, the HMF case features a tower of solitons, each with a different number of nodes. Our results suggest that long-range interactions support solitary waves in a novel manner relative to the short-range case.

More indefinite integrals from Riccati equations

ABSTRACTTwo new methods for obtaining indefinite integrals of a special function using Riccati equations are presented. One method uses quadratic fragments of the Riccati equation, the solutions of which are given by simple quadratic equations. This method is applied to cylinder functions, parabolic cylinder functions and the general solution of the Mathieu equation. No such indefinite integrals for general Mathieu functions seem to have been presented previously. The second method obtains indefinite integrals by assuming simple algebraic forms involving constants for the Riccati variable and then choosing the values of these constants to give simple and interesting integrals. This method is illustrated here for cylinder functions and Associated Legendre functions. All integrals obtained have been checked using Mathematica.

Production performance analysis of fractured vertical wells with SRV in triple media gas reservoirs using elliptical flow

The stimulated reservoir volume (SRV) exists near hydraulic fractures in gas reservoirs after hydraulically fracturing stimulation, which creates one region with distinct gas reservoir properties from initial formation. Due to strong anisotropy, compared with radial flow model, the elliptical flow model is important for the optimal exploitation of triple media gas reservoirs (TMGR) consisting of matrix, vugs and fractures. The paper presents a composite elliptical flow model to analyse the fractured vertical well transient pressure and rate performance in TMGR with SRV. The inter-porosity flow from matrix to fractures is transient flow. Modified Mathieu functions is employed to solve the mathematical model. There are nine flow regimes based on dimensionless pseudo-pressure curves' characteristics and the comparison of transient inter-porosity flow with steady inter-porosity flow is illustrated. Sensitivity analysis for relevant parameters are conducted, such as well radius, SRV radius, mobility ratio, storativity ratio and inter-porosity flow coefficient.

Bidimensional bound states for charged polar nanoparticles

We study the behavior of an electron in the vicinity of a positively charged nanoparticle that has a permanent electric dipole and is of finite size (not point-like) in two dimensions. We place special attention to the fact that a constant of motion, as well as the energy, exists so that the wave functions result from an eigenvalue problem which are classified using two quantum numbers. The (GaAs)3 cluster is used as iconic case. The angular part is solved using Mathieu functions and the radial part by means of a numerical method. The solutions are plotted and the Einstein coefficients are calculated.

A second-order theory for an array of curved wave energy converters in open sea

We present a second-order nonlinear theory for an array of curved flap-type wave energy converters (WECs) in open sea, excited by oblique incident waves. The flap model, when at rest, has a weak displacement of its wetted surface about the vertical plane. Using a perturbation-harmonic expansion technique, we decompose the set of nonlinear governing equations in a sequence of linearized boundary-value problems that can be investigated through analytical methods. The velocity potential at the leading order is solved in terms of elliptical coordinates and related Mathieu functions. Due to quadratic interactions of the first-order potential, a drift flow, a static angular displacement and a bound wave appear at the second order. At the same order the effect of the flap shape forces the first harmonic solution. We then compare an array of flat devices with an array of curved flaps, assuming several shape functions that could be of practical engineering interest. We show that hydrodynamic interactions between the curved flaps and incident waves can have constructive effects in terms of power absorption at large frequencies.

Analysis of Two modified goubau waveguides at THz frequencies: Conical and elliptical structures

In this article, we analyze two modified Goubau waveguides at THz frequencies. The first structure is the conical modified Goubau waveguide, where an air gap has been located between the metal and inner surface of the dielectric layer. The electromagnetic fields have been written in various regions and the propagation loss and characteristic equation have been derived for this waveguide. The cylindrical Goubau waveguide with an elliptical cross-section (or briefly called “Elliptical modified Goubau waveguide”) is the second waveguide that has been analyzed by applying the Mathieu functions in the elliptical coordinates. Both structures have been simulated in HFSS software to validate the theoretical results. There is good agreement between simulation and analytical results. Furthermore, the comparison of modified Goubau structures with typical ones, clearly indicates that better field confinement and low propagation loss can be obtained by utilizing the air-gap in the modified Goubau waveguides. These structures can be utilized in multi-channel communications and designing antenna feed at THz frequencies.

High-Accuracy Calculation of Eigenvalues of the Laplacian in an Ellipse (with Neumann Boundary Condition)

A numerical technique for solving the eigenvalue problem for the Laplacian in an ellipse is described. The results are based on K.I. Babenko’s ideas. In elliptic coordinates, the variables in the Laplace equation for an ellipse are separated and the problem of calculating the eigenvalues is reduced to the study of Mathieu functions. The integral in the variational principle is computed using a global quadrature rule. The minimization of a quadratic functional is reduced to the minimization of a quadratic form, which leads to an algebraic eigenvalue problem.

Theory of chronoamperometry for the catalytic EC′ mechanism at a band electrode: Highly accurate and efficient computation of the Faradaic currents

Rigorous theoretical expressions are derived, for chronoamperometric limiting currents in the case of the catalytic EC′ mechanism at an infinite inlaid band electrode. Transient and steady state currents (per unit length of the electrode) are considered. Identical diffusion coefficients of the members of the redox pair are assumed. By extending the former theory for uncomplicated charge transfers [L. K. Bieniasz, Electrochim. Acta 178 (2015) 25, J. Electroanal. Chem. 760 (2016) 71], the currents are expressed in terms of series expansions involving Mathieu functions. With the help of the formalism obtained, reference current values having at least 20 accurate significant digits are computed. The reference values are further used to develop numerical procedures (written in C++) for calculating the currents efficiently (a single value requiring about 1–103 microseconds on a modern laptop) and with a high accuracy (relative error moduli not exceeding about 10−15). Apart from their utility for the analysis of experimental data, the procedures should be particularly suitable for validating algorithms and software for electroanalytical simulations and modelling.

Multiparticle time-domain analysis of coherent undulator radiation

The physics involving the emission of coherent undulator radiation from a bunch of correlated electrons wiggling inside an undulator has been very well established in the frequency domain. With the aim of describing the process of emission of coherent undulator radiation in time domain while incorporating the effect of hyperbolic field profile of the undulator and various electron beam parameters, we present a time-domain analysis which can serve as a complimentary formulation to the frequency-domain analysis of the coherent undulator radiation. The formulation requires revisiting the equation of motion of an electron moving inside a planar undulator. An electron beam with substantial spread along the direction of the magnetic field will undergo betatron oscillations due to the hyperbolic profile of the undulator fields. We show that the betatron motion can be conveniently expressed in terms of Mathieu functions. The new set of equations of motion have been used to study the effect of betatron motion on the emitted undulator radiation. The single-particle time-domain analysis is then easily extended to multiparticle systems by correctly accounting for the phase-difference between the individual electromagnetic waves emitted by different electrons occupying different phase-space in the bunch.

The Dependence of Resonance Frequency to Landing Angle in Reciprocal Scattering Phenomena of the Waves From an Elliptical Plasma Dielectric Antenna

Scattering of an electromagnetic plane wave in transverse electric mode which is obliquely incident to an elliptical plasma antenna with a dielectric rod is comprehensively investigated. The field solutions of the incident, transmitted, and the scattered waves will be derived in terms of Mathieu and modified Mathieu functions. Using asymptotic expansions of Mathieu functions, the scattering cross section per unit length is obtained. It will be shown that in such an elliptical plasma antenna, the backscattering cross section depends on the incident angle and the bistatic scattering cross section varies with scattering angle significantly. Such effects are not seen in antennas with circular cross section due to the circular symmetry in them. The validation of the presented formulations and numerical computation will be examined with the circular case that is an asymptotic well-known case. The graphs of backscattering cross section, scattering pattern, and their corresponding to the polarized charge densities are presented. The dependence of the backscattering cross section and the scattering pattern upon the density and geometrical dimension of the plasma and dielectric are investigated. In addition, it will be shown that the resonance frequencies, under which the plasma antenna response will be maximum, essentially depend on the incident angle of the incident wave.

Mathieu Windows for Signal Processing

Two new windows for signal processing are introduced in this work. The windows are based on radial Mathieu functions and their performance is compared with well-known ones such as Kaiser, cosh, exponential, Hamming, prolate, and other windows. Most of the results are focused on 1D systems (or windows) and extensions to 2D and multidimensional cases are straightforward. These new Mathieu windows have three adjusting parameters and, by a proper choice of them, it is possible to have a wide span of window shapes that cover most of the existing ones and they might be more appropriate for some applications. An example dealing with the identification of three tones plus normal noise is studied, and the performance of these Mathieu windows is among the best ones. So, they are highly recommended for spectral or harmonic analysis applications. Moreover, due to the high flexibility in adjusting Mathieu windows shapes, these new proposed windows may be applicable for smoothing finite response filters, and related applications as well.

Generation of Mathieu beams using angular pupil modulation**Project supported by the National Natural Science Foundation of China (Grant No. 11674288).

By using an amplitude-type spatial light modulator to load angular spectrum of Mathieu function distribution along a narrow annular pupils, the Durnin’s experimental setup is extended to generate various types of Mathieu beams. As a special type of Mathieu beams, Bessel beams are also generated using this optical setup. Furthermore, the optical morphology of the Mathieu beams family are also presented and analyzed.

Analytical solution for the short-crested wave diffraction by an elliptical cylinder

The diffraction of short-crested waves around an elliptical cylinder is solved using the elliptical coordinates, and an analytical solution is obtained in terms of Mathieu and modified Mathieu functions. Analytical expressions for the pressure, the water run-up and the total wave forces on an elliptical cylinder are also derived in this study. The wave run-up of short-crested waves on an elliptical cylinder is quite different from that of a plane incident wave. The maxima and location of the wave run-up as well as the total wave forces are affected significantly by the incident angle and wave phase. Moreover, the total wave forces induced by the short-crested wave on a circular cylinder is overall smaller than that induced by the plane wave with the same wave number, but in some cases the total wave forces induced by the short-crested waves on an elliptical cylinder may be larger than that induced by the plane wave. Thus, it is necessary to take the short-crested wave forces in to account in the design of an offshore structure with elliptical cross-section.

Analytical solution for converging elliptic shear wave in a bounded transverse isotropic viscoelastic material with nonhomogeneous outer boundary.

Dynamic elastography methods-based on optical, ultrasonic, or magnetic resonance imaging-are being developed for quantitatively mapping the shear viscoelastic properties of biological tissues, which are often altered by disease and injury. These diagnostic imaging methods involve analysis of shear wave motion in order to estimate or reconstruct the tissue's shear viscoelastic properties. Most reconstruction methods to date have assumed isotropic tissue properties. However, application to tissues like skeletal muscle and brain white matter with aligned fibrous structure resulting in local transverse isotropic mechanical properties would benefit from analysis that takes into consideration anisotropy. A theoretical approach is developed for the elliptic shear wave pattern observed in transverse isotropic materials subjected to axisymmetric excitation creating radially converging shear waves normal to the fiber axis. This approach, utilizing Mathieu functions, is enabled via a transformation to an elliptic coordinate system with isotropic properties and a ratio of minor and major axes matching the ratio of shear wavelengths perpendicular and parallel to the plane of isotropy in the transverse isotropic material. The approach is validated via numerical finite element analysis case studies. This strategy of coordinate transformation to equivalent isotropic systems could aid in analysis of other anisotropic tissue structures.

Iterative Scattering by Two PEMC Elliptic Cylinders

Iterative procedure is implemented to derive rigorous solution to the problem of plane electromagnetic wave scattering by couple of perfect electromagnetic conducting (PEMC) elliptic cylinders due co and cross polarized scattered fields among cylinders. The translation addition theorem for Mathieu functions is enforced to compute the higher order scattered fields by single PEMC elliptic cylinder in terms of the other elliptic cylinder coordination system to impose the boundary conditions. The kth co and cross polarized scattered field coefficient expressions are extracted by iteration procedure without using matrix inversion.

Analytical solution for elliptic shear standing wave pattern in a bounded transversely isotropic viscoelastic material

Dynamic elastography methods—based on photonic, ultrasonic, or magnetic resonance imaging—are being developed for quantitatively mapping the shear viscoelastic properties of biological tissues, which are often altered by disease and injury. These diagnostic imaging methods involve analysis of shear wave motion in order to estimate or reconstruct the tissue’s shear viscoelastic properties. Most reconstruction methods to date have assumed isotropic tissue properties. But, application to tissues like skeletal muscle and brain white matter with aligned fibrous structure resulting in local transverse isotropic mechanical properties would benefit from analysis that takes into consideration anisotropy. A theoretical approach is developed for the elliptic shear wave pattern observed in transverse isotropic materials subjected to axisymmetric excitation normal to the fiber axis. This approach, utilizing Mathieu functions, is enabled via a transformation to an elliptic coordinate system with isotropic properties and a ratio of minor and major axes matching the ratio of shear wavelengths perpendicular and parallel to the plane of isotropy in the transverse isotropic material. The approach is validated via finite element analysis cases studies. This strategy of coordinate transformation to equivalent isotropic systems could aid in analysis of other anisotropic tissue structures. (Grant support: NIH AR071162.)

Ground motions around a deep semielliptic canyon with a horizontal edge subjected to incident plane SH waves

We study scattering of antiplane shear waves induced by a deep semielliptic canyon with a horizontal edge. We employ the region-point-matching technique to cope with the problem considered. Through an auxiliary boundary, a part of the circumference of a semiellipse, the whole analyzed region is divided into two subregions. We express the displacement fields in terms of Mathieu functions. We unify two distinct elliptic coordinates via a simple coordinate transformation relation. Integration of the coordinate transformation relation into the region-point-matching technique simplifies the procedure for constructing simultaneous equations. Imposing the continuity conditions and traction-free ones, we obtain the expansion coefficients. Frequency-domain results demonstrate ground motion variability based on several key factors. Ground surface responses under seismic shaking are also simulated in the time domain.

An analysis of the thickness vibration of an unelectroded doubly-rotated quartz circular plate.

The scalar differential equation in the thickness eigendisplacement for the doubly-rotated quartz plates is applied to analyze thickness vibrations of an unelectroded circular plate with free edges at the neighborhood of the pure thickness vibration mode. The scalar differential equation is transformed into an elliptical coordinate system. With the boundary conditions of free edges, the frequencies and the modes are solved in terms of the Mathieu function and the Modified Mathieu function. The results of frequencies of the fundamental harmonic and its third overtone of an AT-cut quartz circular plate by the present approach agree well with the existing theoretical results and the experiment results. The frequencies and modes of an SC-cut quartz circular plate are investigated by the present approach. The frequencies are close to each other when the order of the harmonics is the same. A rotation angle of the symmetric axes of the vibration modes are observed that is dependent on the anisotropic material constants and the order of the harmonics. This approach has potential applications in the design of the doubly-rotated quartz circular resonators.