Separation Of Variables Research Articles

Decomposition solutions and Bäcklund transformations of the B-type and C-type Kadomtsev–Petviashvili equations

This paper introduces a modified formal variable separation approach, showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev–Petviashvili (BKP) equation. Through the application of this approach, we successfully ascertain decomposition solutions, Bäcklund transformations, the Lax pair, and the linear superposition solution associated with the aforementioned equation. Furthermore, we expand the utilization of this technique to the C-type Kadomtsev–Petviashvili (CKP) equation, leading to the derivation of decomposition solutions, Bäcklund transformations, and the Lax pair specific to this equation. The results obtained not only underscore the efficacy of the proposed approach, but also highlight its potential in offering a profound comprehension of integrable behaviors in nonlinear systems. Moreover, this approach demonstrates an efficient framework for establishing interrelations between diverse systems.

Dynamic response of bored piles with gradient defects during low strain integrity test

With the proposal of the gradient ring-soil pile model, the possibility of utilizing the low strain integrity test for identifying gradient defects of the bored piles embedded in unsaturated soil is confirmed. The gradient ring-soil pile model realizes the simulation of soil-pile interaction at the soil-pile interfaces with arbitrary shapes in the horizontal projection. Compared to existing theoretical answers, this paper extends the low strain test theory to the identification of gradient defects. With the adoption of the Laplace transform, differential operator decomposition theory, variable separation method, and impedance function transfer method, a semi-analytical solution is derived. Taking advantage of the semi-analytical solution, a parametric study is conducted to analyze the variation of the defect shapes on the test signal of the bored pile. Based on the findings, a series of methods for identifying gradient defects in engineering applications are summarized. The main findings can be summarized as follows: 1. The soil filled in the gradient necking defects would significantly decrease the magnitudes of reflected signals at these defects, making them more challenging to be identified; 2. If simply modeling the gradient defects as sudden defects, the severity of defects can be underestimated, resulting in the overestimation of pile capacity; 3. The shape of the gradient defect, as well as the length of the gradient defect, has a significant influence on the reflected signal.

Fast QSC solver: tool for systematic study of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 Super-Yang-Mills spectrum

Integrability methods give us access to a number of observables in the planar N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM. Among them, the Quantum Spectral Curve (QSC) governs the spectrum of anomalous dimensions. Low lying states were successfully studied in the past using the QSC. However, with the increased demand for a systematic study of a large number of states for various applications, there is a clear need for a fast QSC solver which can easily access a large number of excited states. Here, we fill this gap by developing a new algorithm and applied it to study all 219 states with the bare dimension ∆0 ≤ 6 in a wide range of couplings.The new algorithm has an improved performance at weak coupling and allows to glue numerics smoothly the available perturbative data, resolving the previous obstruction. Further ∼ 8-fold efficiency gain comes from C++ implementation over the best available Mathematica implementation. We have made the code and the data to be available via a GitHub repository.The method is generalisable for non-local observables as well as for other theories such as deformations of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM and ABJM. It may find applications in the separation of variables and bootstrability approaches to the correlation functions. Some applications to correlators at strong coupling are also presented.

Decoding the essence of two-character Chinese words: Unveiling valence, arousal, concreteness, familiarity, and imageability through word norming.

Investigation of affective and semantic dimensions of words is essential for studying word processing. In this study, we expanded Tse et al.'s (Behav Res Methods 49:1503-1519, 2017; Behav Res Methods 55:4382-4402, 2023) Chinese Lexicon Project by norming five word dimensions (valence, arousal, familiarity, concreteness, and imageability) for over 25,000 two-character Chinese words presented in traditional script. Through regression models that controlled for other variables, we examined the relationships among these dimensions. We included ambiguity, quantified by the standard deviation of the ratings of a given lexical variable across different raters, as separate variables (e.g., valence ambiguity) to explore their connections with other variables. The intensity-ambiguity relationships (i.e., between normed variables and their ambiguities, like valence with valence ambiguity) were also examined. In these analyses with a large pool of words and controlling for other lexical variables, we replicated the asymmetric U-shaped valence-arousal relationship, which was moderated by valence and arousal ambiguities. We also observed a curvilinear relationship between valence and familiarity and between valence and concreteness. Replicating Brainerd et al.'s (J Exp Psychol Gen 150:1476-1499, 2021; J Mem Lang 121:104286, 2021) quadratic intensity-ambiguity relationships, we found that the ambiguity of valence, arousal, concreteness, and imageability decreases as the value of these variables is extremely low or extremely high, although this was not generalized to familiarity. While concreteness and imageability were strongly correlated, they displayed different relationships with arousal, valence, familiarity, and valence ambiguity, suggesting their distinct conceptual nature. These findings further our understanding of the affective and semantic dimensions of two-character Chinese words. The normed values of all these variables can be accessed via https://osf.io/hwkv7 .

3D colored object reconstruction from a single view image through diffusion

In this paper we propose a novel 3D colored object reconstruction method from a single view image. Given a reference image, a conditional diffusion probabilistic model is built to reconstruct both a 3D point cloud shape and the corresponding color features at each point, and then images from arbitrary views can be synthesized using a volume rendering technique. The approach involves several sequential steps. First, the reference RGB image is encoded into separate shape and color latent variables. Then, a shape prediction module predicts reverse geometric noise based on the shape latent variable within the diffusion model. Next, a color prediction module predicts color features for each 3D point using information from the color latent variable. Finally, a volume rendering module transforms the generated colored point cloud into 2D image space, facilitating training based solely on a reference image. Experimental results demonstrate that the proposed method achieves competitive performance on colored 3D shape reconstruction and novel view image synthesis.

Signature of f(R) gravity via Lemaître–Tolman–Bondi inhomogeneous perturbations

We analyze inhomogeneous cosmological models in the local Universe, described by the Lemaître–Tolman–Bondi (LTB) metric and developed using linear perturbation theory on a homogeneous and isotropic Universe background. Focusing on the different evolution of spherical symmetric inhomogeneities, we compare the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}LTB model, in which the cosmological constant Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} is included in the LTB formalism, with inhomogeneous cosmological models based on fR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R\\right) $$\\end{document} modified gravity theories viewed in the Jordan frame. We solve the system of field equations for both inhomogeneous cosmological models adopting the method of separation of variables: we integrate analytically the radial profiles of local perturbations, while their time evolution requires a numerical approach. The main result of the analysis concerns the different radial profiles of local inhomogeneities due to the presence of a non-minimally coupled scalar field in the Jordan frame of fR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R\\right) $$\\end{document} gravity. While radial perturbations follow a power-law in the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}LTB model, Yukawa-like contributions appear in the fR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R\\right) $$\\end{document} theory. Interestingly, this latter peculiar behavior of radial profile is not affected by the choice of the fR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R\\right) $$\\end{document} functional form. The numerical solution of time-dependent perturbations exhibits a non-diverging profile. This work suggests that investigations about local inhomogeneities in the late Universe may allow us to discriminate if the present cosmic acceleration is caused by a cosmological constant term or a modified gravity effect.

Spectral Galerkin mode-matching method for applications in photonics.

Many engineered photonic devices can be decomposed into parts where the material properties are independent of one or more spatial variables. Numerical mode-matching methods are widely used to simulate such photonic devices due to the efficiency gained by treating the separated variables analytically. Existing mode-matching methods based on piecewise polynomials are more accurate than those based on the global Fourier basis or low-order finite difference, finite-element schemes, but they may exhibit numerical instability when a large number of eigenmodes are used. To overcome this difficulty, we introduce the spectral Galerkin mode matching method (SGMM) based on a global piecewise-polynomial basis and a Galerkin method to solve the eigenmodes. It is shown that the numerical eigenmodes of SGMM preserve the pseudo-orthogonality of the analytical eigenmodes. This property leads to linear systems that are typically well-conditioned. Numerical examples indicate that SGMM is more stable than other mode matching methods, and gives reliable results even when a large number of eigenmodes are used.

Integrability of Dirac equations in static spherical space-times

We consider the Dirac equations in static spherically-symmetric space-times, and we present a type of spinor field whose structure allows the separation of elevation angle and radial coordinate in very general situations. We demonstrate that after such a separation of variables the Dirac equations reduce to two equations that can always be integrated, at least in principle. To prove that ours is a fully-working method, we find an explicit exact solution in the special case of the de Sitter universe.

Relativistic massive and massless scalar fields bound to the Bumblebee gravity's black hole

In this letter, we examine massive and massless scalar quasibound states bound in the static spherically symmetric black hole solution of the Einstein-Bumblebee space-time. We start with constructing the governing relativistic fields' dynamical equation, i.e. the Klein-Gordon equation, component by component. With the help of the ansatz of separation of variables, we find the exact solution of the angular part in terms of Spherical Harmonics while the radial exact solution is discovered in terms of the Confluent Heun function. The quantization of the quasibound state is done by applying the polynomial condition of the Confluent Heun function that reveals a complex-valued energy levels expression for the case of massive scalar, where the real part is the scalar particle's energy and the imaginary part represents the quasibound state's decay. And for a massless scalar, a pure imaginary energy levels is obtained. The quasibound states, thus, describe decaying relativistic states bound in the black hole's gravitational potential well. At the end, we investigate the Hawking radiation of the static Bumblebee black hole's horizon by deriving and investigating the radiation distribution function.

A novel approach to model nance drying considering its shrinkage

Air drying is a viable preservation method to boost the accessibility of nance, a seasonal drupe with very short shelf life; however, information about nance drying is scarce, especially from a mass transfer (MT) standpoint. Besides, the lack of simple models considering the stone fraction (radius of inner stone/radius of nance) and the mesocarp shrinkage in drupes has caused most investigations rely on sphere and cube geometries, contributing to the unreliable estimation of MT properties. This study developed a discrete state-space (SS) formulation with analytical solution to describe nance drying which, unlike a variable separation (VS) solution, can consider a variable stone fraction, caused by outer boundary moving while flesh-pit interface remains still. Nance fruits were air dried (60, 70 and 80 °C) to obtain their moisture kinetics and shrinking behavior. Drying data were used to estimate water diffusivity (D) under rigid (VS and SS solutions) and shrinkable (SS and numerical solution) mesocarp assumptions. Nances lost about 60% of their initial volume at the end of drying, while stone fraction increased from 0.44 to 0.62. D values ranged from 5.3 to 6.7 (×10–10) m2/s with SS approach and variable stone fraction. The major source of error in D was not the formulation (SS or VS) when a rigid mesocarp was considered, but the values assigned to the nance radius (length for diffusion) and stone fraction, an initial or average value along process. SS diffusivities did not differ to those estimated with the full numerical solution under variable stone fraction (p>0.05), while VS parameters were overestimated by 44–46 % as a minimum. SS formulation can be applied to hollow cylinders and other geometries with 2D or 3D MT, while able to include non-isotropic shrinkage and evolving external conditions, a problem normally reserved to pure numerical implementations, offering substantial advantages over VS equations.

Q−Relativistic wave equation of the form i∂^q·ψq + mψ_0 = Eψ

In this paper we introduce a q relativistic wave equation of the form i∂^q·ψq + mψ_0 = Eψ. We present the q-spinorial solutions using the method of separated variables in the q-relativistic wave equation. Some comments are mentioned at the end of the paper.

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

Analytical solution for suction-induced seepage flow during suction bucket installation in multilayered soil with anisotropic permeability

Suction-induced seepage flow can significantly reduce the soil resistance during the installation of suction buckets, thereby ensuring their intended penetration depths and the designed in-service capacities. However, the lack of analytical models describing seepage flow behavior in the literature poses a significant challenge, primarily due to the complexity of seepage boundary conditions and the anisotropy and spatial variation of soil permeability. This paper addresses this gap by presenting a novel analytical solution for analyzing suction-induced seepage flow around buckets, with a particular focus on the general multilayered soil profile featuring anisotropic permeability. The method of separation of variables is used initially to derive general solutions, and the final solutions are subsequently obtained by combining continuous conditions with the orthogonality of Bessel functions. The accuracy of the solutions is confirmed through comparisons with the results obtained from finite element analysis. Furthermore, this study discusses the seepage flow behavior in several typical scenarios, including permeability anisotropy, increased permeability within the bucket, and the presence of an overlying low-permeability layer. The analytical solutions presented in this paper provide a rapid and accurate method for the analysis of suction-induced seepage flow during suction-assisted installations across a wide range of complex soil permeability conditions.

An analytical fast field solver for surface permanent magnet co‐axial magnetic gears

AbstractA mathematical model of co‐axial magnetic gears is given for the prediction of magnetic field distribution and the magnetic forces. The modelling approach is based on the separation of variables method (SVM), Sturm–Liouville problem, and spectral analysis. This method is computationally more efficient and has the ability to account for finite permeability of magnetic saliency. An finite element method (FEM) validation is performed for a case study, and the results are observed to be almost matched. The computational time of the analytical model has been almost in the same order of that of the FEM for a specified rotor position (both in a matter of a second or less); however, when 100 number of rotor positions were accounted in the simulation, the computational time of FEM was increased to 82 s, while the computational complexity of SVM was almost unaffected.

Analytical solution to a coupled system including tuned liquid damper and single degree of freedom under free vibration with modal decomposition method

This research focuses on employing a linear analytical approach to transform free surface waves and velocities into mode coordinates, with the aim of investigating the free vibration behavior of a coupled system consisting of a Single Degree of Freedom and a sloshing tank. Through a series of manipulations and simplifications of the coupled equations, a fourth-order ordinary differential equation is derived to showcase the overall response of the system, highlighting the contribution of each odd mode. Key concepts explored include system stability, mode-specific natural periods, establishment of initial boundary conditions, and formulation of the complete system response. The analytical method applied to study Tuned Liquid Dampers, a type of elevated sloshing tank, reveals that in higher modes, the lower frequency aligns with the structural natural frequency, while the higher frequency is approximately n times the structural natural frequency (where n is the odd mode number). This approach also elucidates why the system's response does not exhibit a higher-frequency component in higher modes. The study further investigates concepts such as employing an initial perturbation to excite higher frequencies and the potential for approximating the system through the first mode. Additionally, a numerical model was developed using variable separation and modal decomposition methods to complement and validate the analytical approach. Finally, further verification of the model was performed using the Preismann scheme applied to the relevant equations and the central upwind applied to nonlinear equations.

Analysis of Hydrodynamic Loads by a Vertical Hollow Cylindrical Structure in A Two-Layer Fluid of Finite Depth

This paper aims to study the effect of surge radiation by a vertically hollow cylinder floating in a two-layer fluid of finite depth. Since most of the important characteristics of the oscillating water column are preserved by the hollow cylindrical structure, so the proposed device can be considered as an oscillating water column (OWC) which is one of the important wave energy device. In this two-layer fluid system, the upper layer is bounded above by a free surface and the lower layer is bounded below by a solid impermeable bottom, and the submerged hollow cylindrical structure allows it to oscillate in a surge mode of motion in the upper layer. On the assumption that wave amplitudes are small in comparison to wavelengths, hence the linear water wave theory is applied to formulate boundary value problems by dividing whole domain into 2 sub-domains. The formulated boundary value problems for surge radiated potentials in each sub-domain; we execute to obtain solutions in each sub-domain by using variable separation and matched eigenfunction expansion methods. With the help of obtained surge radiated potentials, we derive the radiation forces by the device in terms of added mass and damping coefficients. A set of influences of different parameters of the device on the added mass and damping coefficients are demonstrated in both surface and internal wave modes of motion. Hydrodynamic coefficients oscillate highly near a particular frequency, which leads to the resonance phenomena. In addition to that, the 3D plots of the free surface elevation in surface and internal wave modes are presented. A density ratio γ = 0.97 has a higher oscillation than a density ratio γ = 0.93, 0.95. High oscillations occur around a particular frequency ω = 3.83 due to the resonance phenomenon in which the waves incoming match with the natural motion of the object. HIGHLIGHTS A floating hollow vertical cylindrical structure is considered in a two-layer fluid system having finite depth The separation of variables and matched eigenfunction expansion methods are utilized to obtain the solutions to the boundary-value problems The diffracted velocity potentials under the surge mode of motion are evaluated in the identified regions The added mass, damping coefficients, free surface and interface elevations in surface and internal wave modes are evaluated with different set of parameters of the device A number of results are presented regarding the effect of parameters of the device on the added mass, damping coefficients, free surface and interface elevations It is observed that the density ratio of the two-layer fluid has significant effect on the added mass, damping coefficient and the elevations GRAPHICAL ABSTRACT

New Multilinear Variable Separation Solutions of the (3 + 1)-Dimensional Burgers Hierarchy

The multilinear variable separation (MLVS) approach has been proven to be very useful in solving (2 + 1)-dimensional integrable systems. Taking the (3 + 1)-dimensional Burgers hierarchy as an example, we extend the MLVS approach to a whole family of (3 + 1)-dimensional Burgers hierarchy. New exact solutions and universal formulas are obtained, which lead to abundant (3 + 1)-dimensional coherent structures. In particular, two ring-type soliton molecules and their interactions are shown in detail. We also generalize the MLVS results of the (3 + 1)-dimensional Jimbo–Miwa (JM) equation and modified JM equation.

Stationary coupled KdV systems and their Stäckel representations

AbstractIn this article, we investigate stationary coupled Korteweg–de Vries (cKdV) systems and prove that every ‐field stationary cKdV system can be written, after a careful reparameterization of jet variables, as a classical separable Stäckel system in different ways. For each of these parameterizations, we present an explicit map between the jet variables and the separation variables of the system. Finally, we show that each pair of Stäckel representations of the same stationary cKdV system, when considered in the phase space extended by Casimir variables, is connected by an appropriate finite‐dimensional Miura map, which leads to an ‐Hamiltonian formulation for the stationary cKdV system.

3D electromagnetic analytical model for radial-flux eddy-current couplers

PurposeThis paper aims to develop a new 3D analytical model in cylindrical coordinates to study radial flux eddy current couplers (RFECC) while considering the magnetic edge and 3D curvature effects, and the field reaction due to the induced currents.Design/methodology/approachThe analytical model is developed by combining two formulations. A magnetic scalar potential formulation in the air and the magnets regions and a current density formulation in the conductive region. The magnetic field and eddy currents expressions are obtained by solving the 3D Maxwell equations in 3D cylindrical coordinates with the variable separation method. The torque expression is derived from the field solution using the Maxwell stress tensor. In addition to 3D magnetic edge effects, the proposed model takes into account the reaction field effect due to the induced currents in the conducting part. To show the accuracy of the developed 3D analytical model, its results are compared to those from the 3D finite element simulation.FindingsThe obtained results prove the accuracy of the new developed 3D analytical model. The comparison of the 3D analytical model with the 2D simulation proves the strong magnetic edge effects impact (in the axial direction) in these devices which must be considered in the modelling. The new analytical model allows the magnetic edge effects consideration without any correction factor and also presents a good compromise between precision and computation time.Practical implicationsThe proposed 3D analytical model presents a considerably reduced computation time compared to 3D finite element simulation which makes it efficient as an accurate design and optimization tool for radial flux eddy current devices.Originality/valueA new analytical model in 3D cylindrical coordinates has been developed to find the electromagnetic torque in radial flux eddy current couplers. This model considers the magnetic edge effects, the 3D curvature effects and the field reaction (without correction factors) while improving the computation time.

Numerical modelling of thermal jet assisted rock cutting with double PDC cutters

AbstractPreconditioning of rock for drilling operations is a potential method to facilitate the mechanical breakage and mitigate the tool wear. This paper numerically investigates one such preconditioning technique, namely, the thermal jet assisted rock cutting. For this end, a numerical method for solving the governing thermo‐mechanical problem is developed and validated. The continuum approach is chosen to describe the rock failure, being based on a damage‐viscoplasticity model with the Drucker–Prager yield surface and the modified Rankine surface as the tensile cut‐off. In the damage part, separate scalar damage variables are employed for tension and compression. The cutters are modelled as rigid bodies and their interaction with the rock is modelled by imposing contact constraints by the forward increment Lagrange multiplier method, which is compatible with the chosen explicit time integration scheme. A damage‐based erosion criterion is applied to remove the contact nodes surrounded by heavily damaged elements. The thermal jets are modelled with a moving external heat flux boundary condition. The global thermo‐mechanical problem is solved with a staggered approach explicitly in time while applying mass scaling to increase the critical time step. The novel 3D numerical simulations involving two cutters demonstrate the capabilities of the method with a special emphasis on cutter‐thermal jet configurations. Therefore, the present method provides a potential tool for the bit design in thermal jet assisted drilling.