Spectral Plane Research Articles

Receptivity of Rayleigh-Taylor instability to acoustic pulses: Theoretical explanation of pulse propagation

Direct numerical simulation (DNS) of Rayleigh-Taylor instability (RTI) has been reported in Sengupta, et al. (2022) [35] using more than 4 billion points with initial time steps of 7.69 ×10−8s. The set-up consists of quiescent heavy (cold) fluid atop lighter (hot) fluid initially separated by an insulated partition. Removal of this partition creates acoustic pulses in the directions normal to the interface, which provides the receptivity route of RTI to transient acoustic pulses spread over several mega-Hz frequencies, far in excess of ultrasonic frequencies. The DNS and its analysis reveal the pulses to be severely attenuated, which cannot be explained by the classical wave equation. In the present research, after demonstrating the features of the DNS results, we report in detail, the theoretical development of a wave equation incorporating losses based on the Navier-Stokes equation without Stokes' hypothesis for a quiescent ambience. Apart from the physical properties of this altered wave equation, the numerical solution of the same is obtained. The governing partial differential equation (PDE) for the propagation of disturbances in the spectral plane, provides the dispersion relation between wavenumber and circular frequency in the dissipative medium accounting for viscous losses. The presented analysis provides physical properties using the global spectral analysis (GSA). This shows the far-field perturbation to propagate either as attenuated waves or strictly in a diffusive manner depending upon the wavenumber. The PDE is solved numerically by a high-accuracy compact scheme and the four-stage, Runge-Kutta scheme for time advancement. The computed solution is shown to match not only with the developed theoretical analysis but also explains the DNS results for RTI at early times when the computed flow field truly represents the quiescent ambience.

Integrability in gravity from Chern-Simons theory

This paper presents a new perspective on integrability in theories of gravity. We show how the stationary, axisymmetric sector of General Relativity can be described by the boundary dynamics of a four-dimensional Chern-Simons theory. This approach shows promise for simplifying solution generating methods in both General Relativity and higher-dimensional supergravity theories. The four-dimensional Chern-Simons theory presented generalises those for flat space integrable models by introducing a space-time dependent branch cut in the spectral plane. We also make contact with twistor space approaches to integrability, showing how the branch cut defects of four-dimensional Chern-Simons theory arise from a discrete reduction of six-dimensional Chern-Simons theory.

Self-similarity and recurrence in stability spectra of near-extreme Stokes waves

We consider steady surface waves in an infinitely deep two-dimensional ideal fluid with potential flow, focusing on high-amplitude waves near the steepest wave with a 120 $^{\circ }$ corner at the crest. The stability of these solutions with respect to coperiodic and subharmonic perturbations is studied, using new matrix-free numerical methods. We provide evidence for a plethora of conjectures on the nature of the instabilities as the steepest wave is approached, especially with regards to the self-similar recurrence of the stability spectrum near the origin of the spectral plane.

Temporal Estimators of Peculiar GRBs

Abstract In our study, we analyze four peculiar Gamma-Ray Bursts (GRBs), namely GRB070724A, GRB070429B, GRB090426, and GRB120804A, all detected by the Swift space telescope. The focus of our investigation lies in evaluating the temporal estimators, such as emission time and spectral lag time in the third Swift GRB catalog sample. Our findings reveal that the calculated temporal estimators offer diverse insights into the classification of the examined GRBs, hinting at the distinctive nature of these events. This suggests that these GRBs can be considered as ‘peculiar’ when compared to standard GRBs in the spectral plane, specifically in the Amati-like relation. Whereas, the spectral lag analysis of the four GRBs increases the evidence to consider these as ‘peculiar’.

Single shot point-diffraction interferometer by a plate beamsplitter

A single shot point-diffraction interferometer (PDI) is proposed in this paper, in which the object beam is split into two copies by means of a beamsplitting module consisting of one plate beamsplitter and one mirror. One of the reflected beams is spatially filtered as a reference beam by one pinhole array in the spectral plane. The split two beams share almost the same path, and by adjusting the inclined angle of the plate beamsplitter, the carrier frequency and legibility of the interferogram can be modulated easily. Moreover, the contrast of the interferogram can be adjusted by changing different plate beamsplitters. Compared to conventional common path off-axis interferometers, this PDI has a simple optical setup, easy optical implementation and outstanding measurement ability with high precision, measurement efficiency and stability. Several experimental results will be provided to demonstrate the effectiveness of the proposed system.

The Spectral Condition, Plane Waves, and Harmonic Analysis in de Sitter and Anti-de Sitter Quantum Field Theories

We review the role of the spectral condition as a characteristic of Minkowski, de Sitter, and anti-de Sitter quantum field theories. We also discuss the role of plane waves which are compatible with the relevant analyticity domains linked to the spectral condition(s) and discuss harmonic analysis in terms of them.

Single line of sight frame camera based on the RadOptic effect of ultrafast semiconductor detector

A new optical beam splitting method is proposed, based on which the optical frame camera capable of capturing multiple frames in a single exposure is designed and experimentally verified. The operation of the frame camera is based on an ultra-fast response semiconductor detector. It is equipped with an optical beam splitter and an optical imaging module. The ultrafast semiconductor detector receives an optical pulse that produces a transient refractive index change, and ultrafast physical processes are recorded by diffracting the probe laser through the transient phase grating. The interaction of an X-ray pulse with a semiconductor detector to produce a phase grating is simulated, based on the Monte Carlo method. The optical beam splitting mode separates a laser into two optical pulses with a certain time difference in the direction of polarization perpendicular to each other. The imaging module filters the diffracted probe laser in the spectral plane and then images multiple frames. The frame camera was used to record the temporal and spatial distribution characteristics of femtosecond laser pulses with a temporal resolution of 4.1 ps. This frame camera has great potential and value for applying to experimental studies of inertial confinement fusion.

The instabilities beyond modulational type in a repulsive Bose–Einstein condensate with a periodic potential

The instabilities of the nontrivial phase elliptic solutions in a repulsive Bose–Einstein condensate (BEC) with a periodic potential are investigated. Based on the defocusing nonlinear Schrödinger (NLS) equation with an elliptic function potential, the well-known modulational instability (MI), the more recently identified high-frequency instability, and an unprecedented – to our knowledge – variant of the MI, the so-called isola instability are identified numerically. Upon varying parameters of the solutions, instability transitions occur through suitable bifurcations, such as the Hamiltonian-Hopf one. Specifically, (i) increasing the elliptic modulus k of the solutions, we find that MI switches to the isola instability and the dominant disturbance has twice the elliptic wave’s period, corresponding to a Floquet exponent μ=π2K(k). The isola instability arises from the collision of spectral elements at the origin of the spectral plane. (ii) Upon varying V0, the transition between MI and the high-frequency instability occurs. Differently from the MI and isola instability where the collisions of eigenvalues happen at the origin, the high-frequency instability arises from pairwise collisions of nonzero, imaginary elements of the stability spectrum; (iii) In the limit of sinusoidal potential, we show that MI occurs from a collision of eigenvalues with μ=π2K(k) at the origin; (iv) we also examine the dynamic byproducts of the instability in chaotic fields generated by its manifestation. An interesting observation is that, in addition to MI, the isola instability could also lead to dark localized events in the scalar defocusing NLS equation.

Effective Non-Iterative Phase Retrieval of 2-D Bandlimited Signals With Applications to Antenna Characterization and Diagnostics

The phase retrieval problem is dealt with for the challenging case where just a single set of (phaseless) field data is available. In particular, the adoption of a new set of intersecting curves in the spectral plane allows decisive improvements over our recent approaches emulating the solution of crosswords puzzles, thus removing relevant limitations on the source size. Numerical examples support the given theory and confirm the effectiveness of the developed procedure.

Dispersion analysis of numerical schemes using 2D compressible linearized Navier–Stokes equation for direct numerical simulation

The dispersion analysis of numerical schemes for finite difference computations based on 2D compressible linearized Navier–Stokes equation (LNSE) is presented here. The analysis presents a more holistic view of the applicability of the schemes while solving compressible NSE, rather than analyzing model 1D or 2D convection or convection–diffusion equations. A theoretical dispersion analysis of model 2D−LNSE is performed first which categorizes it into vortical, entropic and two acoustic modes. The variation of the theoretical dispersion relation for each of the modes with wavenumber, Mach number and Reynolds number is analyzed. It is noted that the entropic mode is the most diffusive among all the modes. Two acoustic modes are less diffusive than the vortical mode up to a certain absolute wavenumber Kb depending upon the Mach number and Reynolds number. It is also shown that while vortical and entropic modes are non-dispersive in nature, the two acoustic modes are dispersive only up to K=Kb. Next, numerical dispersion and diffusion corresponding to each of the modes are analyzed for second order central difference (CD2) and sixth order central compact difference scheme proposed in Lele (J. Comput. Phys., Vol-103(1), 1992) for discretization of convective and diffusive derivatives using fourth-order Runge–Kutta (RK4) time integration method. Results show the existence of numerical unstable zone and regions where spurious upstream propagating waves are noted for each of the modes in the spectral plane. From these and the relative size of the dispersion relation preserving zones for the modes, applicability of the numerical schemes can be assessed. Analysis of the variation of the numerical unstable zone with Mach number, orientation of the mean flow and the unit CFL number Nc=Δt/Δx for each of the modes are presented. It is noted that entropic mode displays the lowest critical CFL number of all the four modes. Lastly, linearized and nonlinear numerical simulations for the convection of acoustic and vortical pulse with varying spectral content are presented. The linearized simulations are also compared with corresponding exact solutions. Obtained results indicate a direct correspondence of the stability analysis performed based on 2D−LNSE even for nonlinear simulations which result in additional intra modal energy transfer and interactions.

On the spectrum of the periodic focusing Zakharov–Shabat operator

The spectrum of the focusing Zakharov–Shabat operator on the circle is studied, and its explicit dependence on the presence of a semiclassical parameter is also considered. Several new results are obtained. In particular: (i) it is proved that the resolvent set is comprised of two connected components; (ii) new bounds on the location of the Floquet and Dirichlet spectra are obtained, some of which depend explicitly on the value of the semiclassical parameter; (iii) it is proved that the spectrum localizes to a “cross” in the spectral plane in the semiclassical limit. The results are illustrated by discussing several examples in which the spectrum is computed analytically or numerically.

Periodic traveling waves in the ϕ^{4} model: Instability, stability, and localized structures.

We consider the instability and stability of periodic stationary solutions to the classical ϕ^{4} equationnumerically. In the superluminal regime, the model possesses dnoidal and cnoidal waves. The former are modulationally unstable and the spectrum forms a figureeight intersecting at the origin of the spectral plane. The latter can be modulationally stable, and the spectrum near the origin in that case is represented by vertical bands along the purely imaginary axis. The instability of the cnoidal states in that case stems from elliptical bands of complex eigenvalues far from the spectral plane origin. In the subluminal regime, there exist only snoidal waves which are modulationally unstable. Considering the subharmonic perturbations, we show that the snoidal waves in the subluminal regime are spectrally unstable with respect to all subharmonic perturbations, while for the dnoidal and cnoidal waves in the superluminal regime, the transition between the spectrally stable state and the spectrally unstable state occurs through a Hamiltonian Hopf bifurcation. The dynamical evolution of the unstable states is also considered, leading to some interesting localization events on the spatio-temporal backgrounds.

Soliton Shielding of the Focusing Nonlinear Schrödinger Equation.

We first consider a deterministic gas of N solitons for the focusing nonlinear Schrödinger (FNLS) equation in the limit N→∞ with a point spectrum chosen to interpolate a given spectral soliton density over a bounded domain of the complex spectral plane. We show that when the domain is a disk and the soliton density is an analytic function, then the corresponding deterministic soliton gas surprisingly yields the one-soliton solution with the point spectrum the center of the disk. We call this effect soliton shielding. We show that this behavior is robust and survives also for a stochastic soliton gas: indeed, when the N-soliton spectrum is chosen as random variables either uniformly distributed on the circle, or chosen according to the statistics of the eigenvalues of the Ginibre random matrix the phenomenon of soliton shielding persists in the limit N→∞. When the domain is an ellipse, the soliton shielding reduces the spectral data to the soliton density concentrating between the foci of the ellipse. The physical solution is asymptotically steplike oscillatory, namely, the initial profile is a periodic elliptic function in the negative x direction while it vanishes exponentially fast in the opposite direction.

Semiclassical WKB problem for the non-self-adjoint Dirac operator with an analytic rapidly oscillating potential

In this paper we examine the semiclassical behavior of the scattering data of a non-self-adjoint Dirac operator with a rapidly oscillating potential that is complex analytic in some neighborhood of the real line. Some of our results are rigorous and quite general. On the other hand, complete and concrete understanding requires the investigation of the WKB geometry of specific examples. For such detailed computations we use a particular example that has been investigated numerically more than 20 years ago by Bronski and Miller and rely heavily on their numerical computations. Mostly employing the exact WKB method, we provide the complete rigorous uniform semiclassical analysis of the Bohr-Sommerfeld condition for the location of the eigenvalues across unions of analytic arcs as well as the associated norming constants. For the reflection coefficient as well as the eigenvalues near 0 in the spectral plane, we employ instead an older theory that has been developed in great detail by Olver. Our analysis is motivated by the need to understand the semiclassical behavior of the focusing cubic NLS equation with initial data Aexp⁡{iS/ϵ}, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator enables the solution of the NLS equation via inverse scattering theory.

A high-order asymptotic analysis of the Benjamin–Feir instability spectrum in arbitrary depth

We investigate the Benjamin–Feir (or modulational) instability of Stokes waves, i.e. small-amplitude, one-dimensional periodic gravity waves of permanent form and constant velocity, in water of finite and infinite depth. We develop a perturbation method to describe to high-order accuracy the unstable spectral elements associated with this instability, obtained by linearizing Euler's equations about the small-amplitude Stokes waves. These unstable elements form a figure-eight curve centred at the origin of the complex spectral plane, which is parametrized by a Floquet exponent. Our asymptotic expansions of this figure-eight are in excellent agreement with numerical computations as well as recent rigorous results by Bertiet al.(Full description of Benjamin–Feir instability of Stokes waves in deep water, 2021, arXiv:2109.11852) and Bertiet al.(Benjamin–Feir instability of Stokes waves in finite depth, 2022, arXiv:2204.00809). From our expansions, we derive high-order estimates for the growth rates of the Benjamin–Feir instability and for the parametrization of the Benjamin–Feir figure-eight curve with respect to the Floquet exponent. We are also able to compare the Benjamin–Feir and high-frequency instability spectra analytically for the first time, revealing three different regimes of the Stokes waves, depending on the predominant instability.

Synthesis of Orbital Angular Momentum Antennas for Target Localization

AbstractThe exploitation of Orbital Angular Momentum (OAM) is proposed to design circular‐support continuous aperture distributions and concentric‐ring arrays for target‐localization applications. The proposed approach allows generating a single isolated null of the field in the target direction, which is of interest with respect to common techniques leading to a line (or perpendicular lines) of power‐pattern zeroes in the spectral plane. In the presented design procedure, the optimal continuous aperture source is first identified by resorting to the most suitable OAM vortex and effective tools such as the Singular Value Decomposition of the involved operator and a smart Convex Programming optimization. Then, by exploiting fast deterministic algorithms, this continuous source is discretized into circular‐ring arrays emulating it at best. Relevant applications including equispaced arrays, sum‐difference reconfigurable arrays, and isophoric sparse arrays are presented. The proposed approach is also compared with the state‐of‐the‐art techniques and assessed through full‐wave electromagnetic simulations.

Empirical mode decomposition and power spectrum filtering for detection of frequency channels related to multi-scale geochemical anomalies: Metal exploration targeting in Moalleman district, NE Iran

Geochemical anomaly patterns for metal exploration targeting usually reveal non-stationarity due mainly to the non-linear synthesis of ore-forming processes. The spectrum–area (S–A) multifractal filtering and the bidimensional empirical mode decomposition (BEMD) are robust methodologies for detecting geochemical anomalies with different scaling properties. The former portrays the generalized scale invariance characteristics of a geochemical image in the two-dimensional Fourier frequency domain. It relies on estimating scaling exponents of power-law functions in the spectral energy density plane to design filter functions that amplify a geochemical signal in specific frequency channels. The BEMD decomposes a geochemical image into a number of bidimensional intrinsic mode functions (BIMFs). It represents the spatial–frequency–energy distribution of a given image that suits non-stationary geochemical signals arising from non-linear Earth systems. This paper gauges the relevance of BEMD and S–A in signalizing frequency channels that correspond to regional- and local-scale geochemical anomalies through a comparative experiment in the Moalleman district, NE Iran. The BEMD filtered the multi-element anomaly map adaptively into a set of BIMFs (BIMF1, BIMF2, and BIMF3) and a residual signal, in order of decreasing frequency, yielding geochemical patterns with scaling characteristics from local to regional. The S–A model filtered the multi-element anomaly map into very high, high, and low frequency channels that characterize noise signatures and geochemical anomalies associated with local and regional ore-forming processes, respectively. The results indicate that the high frequency channel derived from the BEMD filtering (i.e., BIMF1) can portray more effectively local-scale metal-vectoring anomalies distributed at/around ore-controlling faults/fractures. However, the low frequency channel yielded by S–A filtering has enhanced effectively geochemical anomaly architectures that were developed through regional metallogenesis. Therefore, we considered the local-scale geochemical anomalies (i.e., BIMF1) to delimit metal-vectoring targets, whereby four exploration priorities were defined as new areas for further metal prospecting within the study area.

АНАЛИЗ СПЕКТРАЛЬНОЙ ПЛОТНОСТИ: ПРИМЕНЕНИЕ МЕТОДА ПРИ ОПРЕДЕЛЕНИИ ГРАНИЦ ДОЛГОСРОЧНЫХ ЭКОНОМИЧЕСКИХ ЦИКЛОВ

Verification of hypotheses in economic science is a necessary condition for the further development of the put forward scientific assumptions. In this part, the expansion of the set of available means of testing the scientific validity of the revealed regularities and hypotheses built on their basis is an important direction in the development of scientific thought. It is known that one of the directions of economic theory tries to explain economic processes on the basis of physical laws. Taking into account the controversial nature of this approach, a natural question arises concerning the possibility of applying in economic research the methods successfully tested in exact sciences. The presented article describes such an experience, where the method of spectral plane analysis was used to verify the assumption of the existence of systemic economic crises and to determine the chronological boundaries of longterm economic cycles.

On the Use of the Padé–Fourier Approximation in Fast Evaluation of the Green’s Function of Layered Media

Efficient Green’s function (GF) evaluation in layered media is a holy grail of wave theory in general and electromagnetics in particular. While there is a very large amount of knowledge in this context with a vast literature, there are yet challenging cases such as the GF in thick lossy media and the GF in thick media with negative parameters. Here, we propose a technique that can nicely tackle these issues. Our approach is based on a rational function approximation of the spectra using the Fourier–Padé approximation that is carried out in a conformal mapped spectral plane. We show that this approach can be used in challenging scenarios, such as very thick and lossy layers, materials with negative parameters such as in plasmonics, and even to approximate a dominant branch-cut contribution far from the source.

Hyperspectral Image Reconstruction Based on Spatial-Spectral Domains Low-Rank Sparse Representation

The enormous amount of data that are generated by hyperspectral remote sensing images (HSI) combined with the spatial channel’s limited and fragile bandwidth creates serious transmission, storage, and application challenges. HSI reconstruction based on compressed sensing has become a frontier area, and its effectiveness depends heavily on the exploitation and sparse representation of HSI information correlation. In this paper, we propose a low-rank sparse constrained HSI reconstruction model (LRCoSM) that is based on joint spatial-spectral HSI sparseness. In the spectral dimension, a spectral domain sparsity measure and the representation of the joint spectral dimensional plane are proposed for the first time. A Gaussian mixture model (GMM) that is based on unsupervised adaptive parameter learning of external datasets is used to cluster similar patches of joint spectral plane features, capturing the correlation of HSI spectral dimensional non-local structure image patches while performing low-rank decomposition of clustered similar patches to extract feature information, effectively improving the ability of low-rank approximate sparse representation of spectral dimensional similar patches. In the spatial dimension, local-nonlocal HSI similarity is explored to refine sparse prior constraints. Spectral and spatial dimension sparse constraints improve HSI reconstruction quality. Experimental results that are based on various sampling rates on four publicly available datasets show that the proposed algorithm can obtain high-quality reconstructed PSNR and FSIM values and effectively maintain the spectral curves for few-band datasets compared with six currently popular reconstruction algorithms, and the proposed algorithm has strong robustness and generalization ability at different sampling rates and on other datasets.