Spectral Equation Research Articles

A spectral domain integral equation technique for rough surface scattering problems

ABSTRACT In this paper, a novel formulation for scattering from one-dimensional (1D) rough surface problem has been presented. The formulation is based on the representation of the scattering field in spectral domain with an unknown spectral coefficient which is solved by applying Taylor series and Fourier Transforms techniques. It is also shown that forward–backward (FB) acceleration technique can be adapted to the spectral formulation to improve the effectiveness in a limited region of convergence. The obtained results are compared with the conventional method of moments (MoM) and the limitations of the formulation are analyzed in detail.

Solving fractional Laplacian visco-acoustic wave equations on complex-geometry domains using Grünwald-formula based radial basis collocation method

We propose a radial basis function collocation method (RBF method) to solve fractional Laplacian visco-acoustic wave equation for the Earth media having heterogeneous velocity model and complex geometry. Unlike the fractional Laplacian wave equation proposed in Zhu and Harris (2014), the wave equation we consider has a different definition for the fractional Laplacian. Specifically, spectral and Riesz fractional Laplacians are considered in Zhu and Harris (2014) and the present paper, respectively. Accordingly, the Fourier pseudospectral method (FPS method) and the RBF method are employed to solve the spectral and the Riesz fractional Laplacian wave equations. The two wave equations are observed to produce obviously different wavefields. We demonstrate the validity and flexibility of the proposed RBF method by considering five benchmarks of seismic forward modeling: (1) two-dimensional Earth media with four types of velocity models (homogeneous, two-layer, homogeneous but complex-geometry, and heterogeneous models) and (2) a spherical medium with homogeneous velocity model. We make a three-way comparison among numerical solutions to the Riesz fractional Laplacian, the spectral fractional Laplacian, and the integer-order visco-acoustic wave equations, and observe that when wave attenuation is weak the Riesz wave equation yields more similar wavefield to that of the integer-order wave equation than the spectral wave equation does. Furthermore, uniform and quasi-uniform layouts for collocation points of the RBF method are considered, and the latter layout turns out to be economical since it can preserve the solution accuracy with the minimum number of collocation points. The RBF method is truly mesh-free and dimension-free and can easily handle high-dimensional, irregular domains. Additionally, the method is easier to implement than element-based methods, such as finite element and spectral element methods, for discretizing the Riesz fractional Laplacian.

Higher-order interactional solutions and rogue wave pairs for the coupled Lakshmanan–Porsezian–Daniel equations

The N-fold Darboux transformation of the coupled Lakshmanan–Porsezian–Daniel (LPD) equations is constructed. Based on the Darboux transformation and the limiting technique, we investigate two kinds of solutions for the coupled LPD equations, which are higher-order interactional solutions and rogue wave (RW) pairs. Through considering the double-root situation of the spectral characteristic equation for the matrix in the Lax pair, we give the higher-order interactional solutions among higher-order RWs, multi-bright (dark) solitons and multi-breather. Besides, we consider the triple-root situation of the spectral characteristic equation and get the higher-order RW pairs. It demonstrates that the RW pairs are greatly different form the traditional higher-order RWs. The fist-order RW pairs can split into two traditional first-order RWs, and four or six traditional fundamental RWs can emerge from the second-order case. The corresponding dynamics of these explicit solutions are discussed in detail.

Enhanced band-gap properties of an acoustic metamaterial beam with periodically variable cross-sections

In order to enhance the vibration reduction capacity in a wide frequency range, a novel acoustic metamaterial beam with periodically variable cross-sections is designed by combining the mechanisms of the Bragg scattering and the locally resonant band-gaps. The spectral element method (SEM) is extended to investigate the band-gap properties of the novel acoustic metamaterial beam. Based on the exact frequency responses obtained by solving the spectral equations in the frequency domain, it is found that the interaction of the Bragg scattering and the locally resonant band-gaps can enlarge the band-gap widths. More band-gaps can be achieved if the material periodicity is harmonized with the cross-section periodicity. When these two kinds of band-gaps exist simultaneously, the locations of the locally resonant band-gaps do not strictly correspond to the resonator's natural frequencies. The width and the attenuation ability of the Bragg scattering band-gaps become wider and stronger with the increase of the ratio of the cross-sectional sizes of the two sub-cells. The experimentally measured band-gap positions and widths agree very well with the SEM results, which guarantees the correctness of the numerical results.

Proposal for a New Quantum Theory of Gravity III: Equations for Quantum Gravity, and the Origin of Spontaneous Localisation

Abstract We present a new, falsifiable quantum theory of gravity, which we name non-commutative matter-gravity. The commutative limit of the theory is classical general relativity. In the first two papers of this series, we have introduced the concept of an atom of space-time-matter (STM), which is described by the spectral action in non-commutative geometry, corresponding to a classical theory of gravity. We used the Connes time parameter, along with the spectral action, to incorporate gravity into trace dynamics. We then derived the spectral equation of motion for the gravity part of the STM atom, which turns out to be the Dirac equation on a non-commutative space. In the present work, we propose how to include the matter (fermionic) part and give a simple action principle for the STM atom. This leads to the equations for a quantum theory of gravity, and also to an explanation for the origin of spontaneous localisation from quantum gravity. We use spontaneous localisation to arrive at the action for classical general relativity (including matter source) from the action for STM atoms.

Rationally weighted Hurwitz numbers, Meijer G-functions and matrix integrals

The quantum spectral curve equation associated with KP τ-functions of hypergeometric type serving as generating functions for rationally weighted Hurwitz numbers is solved by generalized hypergeometric series. The basis elements spanning the corresponding Sato Grassmannian element are shown to be Meijer G-functions, or their asymptotic series. Using their Mellin integral representation, the τ-function, evaluated at the trace invariants of an externally coupled matrix, is expressed as a matrix integral.

Analytic approach to calculations of mass spectra and decay constants of heavy-light quarkonia in the framework of Bethe-Salpeter equation

This work is an extension of the work in [Phys. Rev. D 97, 034021 (2018)] to ground and excited states of ${0}^{++}$, ${0}^{\ensuremath{-}+}$, and ${1}^{\ensuremath{-}\ensuremath{-}}$ of heavy-light ($c\overline{u},c\overline{s},b\overline{u},b\overline{s}$, and $b\overline{c}$) quarkonia in the framework of a QCD motivated Bethe-Salpeter equation (BSE) by making use of the exact treatment of the spin structure (${\ensuremath{\gamma}}_{\ensuremath{\mu}}\ensuremath{\bigotimes}{\ensuremath{\gamma}}_{\ensuremath{\mu}}$) in the interaction kernel, in contrast to the approximate treatment of the same in our previous works [H. Negash and S. Bhatnagar, Int. J. Mod. Phys. E 25, 1650059 (2016)., S. Bhatnagar and L. Alemu, Phys. Rev. D 97, 034021 (2018).]). In this $4\ifmmode\times\else\texttimes\fi{}4$ BSE framework, the coupled Salpeter equations for $Q\overline{q}$ (that are more involved than the equal mass ($Q\overline{Q}$) mesons) are first shown to decouple for the confining part of interaction, under heavy-quark approximation, and analytically solved, and later the one-gluon-exchange interaction is perturbatively incorporated, leading to their mass spectral equations. The analytic forms of wave functions obtained from these equations are then used for calculation of leptonic decay constants of ground and excited states of ${0}^{\ensuremath{-}+}$ and ${1}^{\ensuremath{-}\ensuremath{-}}$ as a test of these wave functions and the over all framework.

Reproducing kernel for elastic Herglotz functions

We study the elastic Herglotz wave functions, which are entire solutions of the spectral Navier equation appearing in linearized elasticity theory with $$L^2$$-far-field patterns. We characterize in three-dimensions the set of these functions $${\varvec{\mathcal {W}}},$$ as a closed subspace of a Hilbert space $${\varvec{\mathcal {H}}}$$ of vector-valued functions such that they and their spherical gradients belong to a certain weighted $$L^2$$ space. This allows us to prove that $${\varvec{\mathcal {W}}}$$ is a reproducing kernel Hilbert space and to calculate the reproducing kernel. Finally, we outline the proof for the two-dimensional case and give the corresponding reproducing kernel.

Plasmon-Polariton from a Helical State in a Dirac Magnet

Electromagnetic field interacting with a topologically protected one-dimensional electron helical state is shown to support a one-dimensional plasmon-polariton, which is a collective electron excitation dressed with the electromagnetic field. The electronic helical state arises at the surface of three-dimensional topological insulator in the proximity of the ferromagnet and is localized at the magnetization domain wall. This opens the possibilities to manipulate quantum optical states by altering magnetic domain configurations. An exact spectral equation for such topological plasmon-polariton is derived.

On separation of variables for reflection algebras

We implement our new separation of variables (SoV) approach for open quantum integrable models associated to higher rank representations of the reflection algebras. We construct the (SoV) basis for the fundamental representations of the Y(gln) reflection algebra associated to general integrable boundary conditions. Moreover, we give the conditions on the boundary K-matrices allowing for the transfer matrix to be diagonalizable with simple spectrum. These SoV basis are then used to completely characterize the transfer matrix spectrum for the rank one and two reflection algebras. The rank one case is developed for both the rational and trigonometric fundamental representations of the 6-vertex reflection algebra. On the one hand, we extend the complete spectrum characterization to representations previously lying outside the SoV approach, e.g. those for which the standard algebraic Bethe Ansatz applies. On the other hand, we show that our new SoV construction can be reduced to the generalized Sklyanin’s one whenever it is applicable. The rank two case is developed explicitly for the fundamental representations of the Y(gl3) reflection algebra associated to general integrable boundary conditions. For both rank one and two our SoV approach leads to a complete characterization of the transfer matrix spectrum in terms of a set of polynomial solutions to the corresponding quantum spectral curve equation. Those are finite difference functional equations of order equal to the rank plus one, i.e. here two and three respectively for the Y(gl2) and Y(gl3) reflection algebras.

Effective Hamiltonian for QCD on the Light Front

A semiphenomenological model for the effective Hamiltonian of Quantum Chromodynamics on the light front in (3 + 1)-dimensional space-time is proposed and the spectral equation for the mass squared of quark-antiquark states is considered. Proposed Hamiltonian includes a term dependent on the zero mode of the gluon field, which is introduced as an independent dynamic variable. The inclusion of this term allows to obtain quark and antiquark confinement in transverse directions in the Light Front coordinates. We can obtain confinement also in the longitudinal coordinates taking into account the four-fermion term in the Hamiltonian.

Proposal for a New Quantum Theory of Gravity II

Abstract In the first article of this series, we have introduced the concept of an atom of space-time-matter (STM), which is described by the spectral action of noncommutative geometry, corresponding to a classical theory of gravity. In the present work, we use the Connes time parameter, along with the spectral action, to incorporate gravity into trace dynamics. We then derive the spectral equation of motion for the STM atom, which turns out to be the Dirac equation on a noncommutative space.

On the differential equation for the Laguerre–Sobolev polynomials

The Laguerre–Sobolev polynomials form an orthogonal polynomial system with respect to a Sobolev-type inner product associated with the Laguerre measure on the positive half-axis and two point masses M,N>0 at the origin involving functions and derivatives. These polynomials have attracted much interest over the last two decades, since they became known to satisfy, for any value of the Laguerre parameter α∈N0, a spectral differential equation of finite order 4α+10. In this paper we establish a new explicit representation of the corresponding differential operator which consists of a number of elementary components depending on α,M,N. Their interaction reveals a rich structure both being useful for applications and as a model for further investigations in the field. In particular, the Laguerre–Sobolev differential operator is shown to be symmetric with respect to the Sobolev-type inner product.

Complete spectrum of quantum integrable lattice models associated to by separation of variables

In this paper we apply our new separation of variables approach to completely characterize the transfer matrix spectrum for quantum integrable lattice models associated to fundamental evaluation representations of with quasi-periodic boundary conditions. We consider here the case of generic deformations associated to a parameter q which is not a root of unity. The separation of variables (SoV) basis for the transfer matrix spectral problem is generated by using the action of the transfer matrix itself on a generic co-vector of the Hilbert space, following the general procedure described in our paper (Maillet and Niccoli 2018 J. Math. Phys. 59 091417). Such a SoV construction allows to prove that for general values of the parameters defining the model the transfer matrix is diagonalizable and with simple spectrum for any twist matrix which is also diagonalizable with simple spectrum. Then, using together the knowledge of such a SoV basis and of the fusion relations satisfied by the hierarchy of transfer matrices, we derive a complete characterization of the transfer matrix eigenvalues and eigenvectors as solutions of a system of polynomial equations of order n + 1. Moreover, we show that such a SoV discrete spectrum characterization is equivalently reformulated in terms of a finite difference functional equation, the quantum spectral curve equation, under a proper choice of the set of its solutions. A construction of the associated Q-operator induced by our SoV approach is also presented.

A numerical study of oil spill prediction in the Gulf of Thailand using ocean wave model

Oil spill is the most important cause of marine pollution. It affects marine ecosystem, an economic system, and human society. The main chemical and physical properties are the so-called oil weathering processes; spreading is one of the processes. In this research, the spreading is mainly considered because it is the most dominant process at the beginning of an oil spill incident. A mathematical model for oil prediction, based on Fay’s hypothesis and Lehr’s formula, is used to simulate the spreading of oil spill. The model needs wind speed as the main factor. In this paper, the wave model, namely WAM, is used for simulation of atmospheric and oceanographic states. The wave model is based on the wave spectral balance equation. The numerical results from the oil spill model based on ocean wave modeling are shown to study oil spill incident at three different locations in the Gulf of Thailand.

A new spectral integral equation method for solving two-dimensional unsteady advection-diffusion equations via Chebyshev polynomials

PurposeThe purpose of this paper is to develop an efficient and reliable spectral integral equation method for solving two-dimensional unsteady advection-diffusion equations.Design/methodology/approachIn this study, the considered two-dimensional unsteady advection-diffusion equations are transformed into the equivalent partial integro-differential equations via integrating from the considered unsteady advection-diffusion equation. After this stage, by using Chebyshev polynomials of the first kind and the operational matrix of integration, the integral equation would be transformed into the system of linear algebraic equations. Robustness and efficiency of the proposed method were illustrated by six numerical simulations experimentally. The numerical results confirm that the method is efficient, highly accurate, fast and stable for solving two-dimensional unsteady advection-diffusion equations.FindingsThe proposed method can solve the equations with discontinuity near the boundaries, the advection-dominated equations and the equations in irregular domains. One of the numerical test problems designed specially to evaluate the performance of the proposed method for discontinuity near boundaries.Originality/valueThis study extends the intention of one dimensional Chebyshev approximate approaches (Yuksel and Sezer, 2013; Yuksel et al., 2015) for two-dimensional unsteady advection-diffusion problems and the basic intention of our suggested method is quite different from the approaches for hyperbolic problems (Bulbul and Sezer, 2011).

Complete spectrum of quantum integrable lattice models associated to Y(gl(n)) by separation of variables

We apply our new approach of quantum Separation of Variables (SoV) to the complete characterization of the transfer matrix spectrum of quantum integrable lattice models associated to \bm{gl_n}𝐠𝐥𝐧-invariant \bm{R}𝐑-matrices in the fundamental representations. We consider lattices with \bm{N}𝐍-sites and general quasi-periodic boundary conditions associated to an arbitrary twist matrix \bm{K}𝐊 having simple spectrum (but not necessarily diagonalizable). In our approach the SoV basis is constructed in an universal manner starting from the direct use of the conserved charges of the models, e.g. from the commuting family of transfer matrices. Using the integrable structure of the models, incarnated in the hierarchy of transfer matrices fusion relations, we prove that our SoV basis indeed separates the spectrum of the corresponding transfer matrices. Moreover, the combined use of the fusion rules, of the known analytic properties of the transfer matrices and of the SoV basis allows us to obtain the complete characterization of the transfer matrix spectrum and to prove its simplicity. Any transfer matrix eigenvalue is completely characterized as a solution of a so-called quantum spectral curve equation that we obtain as a difference functional equation of order \bm{n}𝐧. Namely, any eigenvalue satisfies this equation and any solution of this equation having prescribed properties that we give leads to an eigenvalue. We construct the associated eigenvector, unique up to normalization, of the transfer matrices by computing its decomposition on the SoV basis that is of a factorized form written in terms of the powers of the corresponding eigenvalues. Finally, if the twist matrix \bm{K}𝐊 is diagonalizable with simple spectrum we prove that the transfer matrix is also diagonalizable with simple spectrum. In that case, we give a construction of the Baxter \bm{Q}𝐐-operator and show that it satisfies a \bm{T}𝐓-\bm{Q}𝐐 equation of order \bm{n}𝐧, the quantum spectral curve equation, involving the hierarchy of the fused transfer matrices.

Evolution of a pair of random inhomogeneous wave systems over infinite-depth water

A system of two coupled nonlinear spectral transport equations is derived for two obliquely interacting narrowband Gaussian random surface wavetrains, slowly varying in space and time. Using these two equations, stability analysis is performed for two initially homogeneous wave spectra, subject to unidirectional perturbations. We observe that the effect of randomness produces a decrease in the growth rate of instability, but it is higher than the growth for a single wavetrain. The growth rate of instability is observed to decrease with the increase in spectral width. doi:10.1017/S144618111900004X

Effect of pressure-dilatation on energy spectrum evolution in compressible turbulence

The effect of internal-kinetic energy exchange on transient spectral energy transfer in compressible turbulence is investigated. We derive the spectral evolution equations for kinetic energy and pressure fields to highlight the key mechanisms that affect the turbulence spectral evolution. Direct numerical simulations of decaying isotropic turbulence are performed from solenoidal, dilatational, and mixed velocity initial conditions. It is shown that internal-kinetic energy exchange arising due to pressure-dilatation renders the dilatational kinetic energy amplitudes at large scales of motion to be oscillatory. The oscillatory behavior of amplitude diminishes with increasing wavenumber. The dilatational energy spectrum also exhibits a wider range of scales due to its inherent tendency to form shocks. The findings are expected to lead to an improved understanding of energy dynamics in high-speed compressible flows.

EVOLUTION OF A PAIR OF RANDOM INHOMOGENEOUS WAVE SYSTEMS OVER INFINITE-DEPTH WATER

A system of two coupled nonlinear spectral transport equations is derived for two obliquely interacting narrowband Gaussian random surface wavetrains, slowly varying in space and time. Using these two equations, stability analysis is performed for two initially homogeneous wave spectra, subject to unidirectional perturbations. We observe that the effect of randomness produces a decrease in the growth rate of instability, but it is higher than the growth for a single wavetrain. The growth rate of instability is observed to decrease with the increase in spectral width.