Spectral Problem Research Articles

Integration of the Korteweg–de Vries type equations with a loaded term in the class of periodic functions

In this paper, we study the integrability of a Korteweg–de Vries type equation with a loaded term in the class of periodic functions using direct and inverse spectral problems posed for the Sturm–Liouville operator on the entire axis. We present some information about the Sturm–Liouville operator with a periodic coefficient and its application to solving a loaded Korteweg–de Vries type equation. We show that the function constructed using the obtained system of Dubrovin differential equations and trace formulas is a solution to the problem posed. We present important consequences about the period of the solution with respect to $x$ and about the analyticity of the solution with respect to $x$ for a Korteweg–de Vries type equation with a loaded term.

On degenerate (p,q)$(p,q)$‐Laplace equations corresponding to an inverse spectral problem

On degenerate (p,q)$(p,q)$‐Laplace equations corresponding to an inverse spectral problem

Convergence from the discrete to the continuous non-linear Fourier transform

In this note, we study the convergence from the discrete to the continuous non-linear Fourier transform. Relations between spectral problems and questions in complex function theory provide a new approach to the study of scattering problems and the non-linear Fourier transform (see A. Poltoratski [Ann. of Math. (2) 199 (2024), pp. 741–793]). In particular, the non-linear Fourier transform can be viewed from the perspective of spectral problems for differential operators. Results of Makarov and Poltoratski [J. Lond. Math. Soc. 108 (2023), pp. 916–977] and Poltoratski and Zhang [J. Funct. Ana. 284 (2023), 29 pp.] can be seen as results for the non-linear Fourier transform. These results are similar to some convergence problems for the discrete non-linear Fourier transform considered by Tao [An introduction to the nonlinear Fourier transform, 2002] and Tao and Thiele [Nonlinear Fourier analysis, 2012].

Solution of boundary value problems for the heat equation with a piecewise constant coefficient using the Fourier method

This paper considers some initial boundary value problems for the heat equation in a bounded segment with a piecewise constant coefficient. Using the method of separation of variables, the problem under consideration is reduced to a spectral problem and eigenvalues and eigenfunctions of the resulting spectral problem are found. It is shown that the system of eigenfunctions forms a Riesz basis. Next, we prove a theorem on the existence and uniqueness of solutions to the initial-boundary value problems under consideration.

Two-phase heat conduction problems with Sturm-type boundary conditions

In this work, the solution is justified using the method of separation of variables for initial-boundary value problems of the heat conduction equation with piecewise constant thermal conductivity coefficient under Sturm-type boundary conditions (separated boundary conditions). Various cases are considered. A system of eigenfunctions of the corresponding spectral problem is constructed and it is proved that this system of eigenfunctions forms a Riesz basis. The theorem of the existence of a unique classical solution to the problem is proved.

Homogenization of nonlocal spectral problems

Homogenization of nonlocal spectral problems

Spectral analysis of eigenvalue problem with the (p(x),q(x))-Laplacian

In this study, a spectral problem that involves the Robin (a(x),b(x))-Laplacian on a bounded domain in Rn is analyzed. With the help of the Ljusternik-Schnirelmann principle, it is established that an increasing sequence of eigenvalues exists. Using the min-max principle, we also established that the infimum of the spectrum is zero, and all the eigenvalues are non-negative. It was further proved that there exists a principal eigenvalue for this problem and that the spectrum is open.

Dirac points and inverse problems of quantum graphs associated with Archimedean tilings

Abstract One interesting phenomenon of graphene is the presence of the conical singularity or Dirac points. Using the quantum graph model, we show that there exist three classes of possible Dirac points for all of the periodic quantum graphs associated with Archimedean tilings, when the potentials are identical and even. They occur at the periodic eigenvalues, anti-periodic eigenvalues, and other double eigenvalues of the dispersion relations, respectively. We also characterize their associated potentials. Moreover, we show that there are no other possible Dirac points. Our work is systematic and rigorous. Finally, we solve an inverse spectral problem for the potential, given the knowledge of the pure point and absolutely continuous spectra.

Spectral Partition Problems with Volume and Inclusion Constraints

Spectral Partition Problems with Volume and Inclusion Constraints

On the Atomic Structure

In recent years, it has been recognized that treatment of the spectral problem should be associated with closed orbits. This paper proves that the closed orbit obtained by Bohr using numerical algebraic equation is reasonable both mathematically and physically and that the de Broglie wavelength of the electron in the hydrogen atom is equal to the circumference of its orbit. By Bohr model and de Broglie’s matter-wave thought, some quantization conditions, electronic transition formulae, a mechanism and details of emitting photon wave train, a filling order of electrons in the periodic table, and Madelung rule have been deduced theoretically. We solve the Löwdin’s challenge problem.

Jade Identification Using Ultraviolet Spectroscopy Based on the SpectraViT Model Incorporating CNN and Transformer

Jade is a highly valuable and diverse gemstone, and its spectral characteristics can be used to identify its quality and type. We propose a jade ultraviolet (UV) spectrum recognition model based on deep learning, called SpectraViT, aiming to improve the accuracy and efficiency of jade identification. The algorithm combines residual modules to extract local features and transformers to capture global dependencies of jade’s UV spectrum, and finally classifying jade using fully connected layers. Experiments were conducted on a UV spectrum dataset containing four types of jade (natural diamond, cultivated diamond (CVD/HPHT), and moissanite). The results show that the algorithm can effectively identify different types of jade, achieving an accuracy of 99.24%, surpassing traditional algorithms based on Support Vector Machines (SVM) and Partial Least Squares Discriminant Analysis (PLS_DA), as well as other deep learning methods. This paper also provides a reference solution for other spectral analysis problems.

Analysis of the three‐dimensional elasticity theory problem for a radially inhomogeneous cylinder

AbstractIn this paper an axisymmetric problem of elasticity theory for a radially inhomogeneous cylinder of small thickness is studied. It is assumed that homogeneous mixed boundary conditions are specified on lateral surfaces of the cylinder, and boundary conditions that leave the cylinder in equilibrium are specified on the ends of the cylinder. It is considered that elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thinness of the cylinder walls. To determine its solution, the method of asymptotic integration is used, based on three iteration processes. All solutions of the equilibrium equation are constructed, satisfying the specified homogeneous boundary conditions on lateral surfaces. It is obtained that the solution of the problem consists of a penetrating solution and a solution of the nature of the boundary layer. An analysis of stress‐strain states corresponding to different types of solutions is carried out. Based on the constructed asymptotic solutions, a connection is found between the solutions obtained by the equation of elasticity theory and by the applied theory of shells. It is shown that the penetrating solution determines the internal stress‐strain state of a radially inhomogeneous cylinder. The stress state determined by the penetrating solution is equivalent to the principal vector of stress acting in a cross section perpendicular to the axis of the cylinder.The first terms of asymptotic expansions of the penetrating solution in a small parameter coincide with solutions in the applied theory of shells. New classes of solutions are defined as the character of a boundary layer, which are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in the applied theories of shells. The first terms of the asymptotic expansion of the solution, having the nature of a boundary layer, are equivalent to the Saint‐Venant edge effect in the theory of heterogeneous plates. Asymptotic formulas for displacement and stresses are constructed, allowing one to calculate the three‐dimensional stress‐strain state of a radially heterogeneous cylinder of small thickness with any predetermined accuracy. Based on the obtained asymptotic expansions, one can estimate the applicability areas of applied theories and construct a refined applied theory for radially heterogeneous cylindrical shells.

Spectral Unmixing of Pigments on Surface of Painted Artefacts Considering Spectral Variability

Abstract. Painted artefacts, such as murals and paintings, are the treasures of human civilization. Pigment is an important component of their surfaces. It is crucial to study the composition and proportion of pigments on the surface of painted artefacts for the field of heritage conservation. In this study, hyperspectral remote sensing was used to invert the abundance of mixed pigments by spectral unmixing. Spectral variability effects are often present in hyperspectral images. Hyperspectral images of cultural artefacts also suffer from spectral variability due to factors such as particle size and impurities within the pigment, and acquisition conditions. Therefore, spectral variability was incorporated into the Linear Mixed Model of the pigment spectral unmixing. The unmixing methods are classified into two categories based on whether or not spectral libraries are used. In the experiment, computer synthesized data and laboratory-made sample blocks of mixed pigments, which mixed with different ratios of Azurite and Malachite, were selected as validation data. Five commonly used algorithms for solving the spectral variability problem, namely MESMA, Fractional Sparse SU, ELMM, RUSAL, and BCM, are used for pigment unmixing and compared with FCLS, which does not consider spectral variability. The results show that ELMM has the highest unmixing accuracy of aRMSE and xRMSE compared to other methods. At the same time, it can be seen from the metric of SAM that ELMM is able to extract reliable endmember variability spectral, which is more suitable for solving the problem of spectral variability in pigment unmixing. Finally, we apply ELMM to real artefacts Yungang Grottoes mural paintings, and obtain a lower xRMSE than FCLS, which improves the unmixing accuracy.

Physics-informed deep learning for structural dynamics under moving load

Physics-informed deep learning has emerged as a promising approach that incorporates physical constraints into the model, reduces the amount of data required, and demonstrates robustness and potential in dealing with limited datasets for a variety of studies. However, several key challenges still exist, with one being the spectral bias problem of deep learning in the simulation of functions with multi-frequency features. To overcome the challenge, this study proposes a novel physics-informed deep learning method, which integrates physics-informed neural network with Fourier transform so as to solve partial differential equations in the frequency domain, thus alleviating the problem of spectral bias of neural networks in the simulation of multi-frequency functions. In addition, the proposed method is used to focus on the forward simulation and parameter inverse identification issues in structural dynamics under moving loads. To illustrate the superiority of the method, the issues of dynamic response of simply supported beams under moving loads are presented as case studies, and the performance of the method in multiple cases is analysed and discussed. The research results demonstrate the feasibility and effectiveness of the method for structural dynamics simulation and parameter inverse identifications using limited datasets.

Sinc method in spectrum completion and inverse Sturm–Liouville problems

Cardinal series representations for solutions of the Sturm–Liouville equation with a complex‐valued potential are obtained, by using the corresponding transmutation operator. Consequently, partial sums of the series approximate the solutions uniformly with respect to in any strip of the complex plane. This property of the obtained series representations leads to their applications in a variety of spectral problems. In particular, we show their applicability to the spectrum completion problem, consisting in computing large sets of the eigenvalues from a reduced finite set of known eigenvalues, without any information on the potential as well as on the constants from boundary conditions. Among other applications this leads to an efficient numerical method for computing a Weyl function from two finite sets of the eigenvalues. This possibility is explored in the present work and illustrated by numerical tests. Finally, based on the cardinal series representations obtained, we develop a method for the numerical solution of the inverse two‐spectra Sturm–Liouville problem and show its numerical efficiency.

On the quadratic closeness of eigenfunctions of «unperturbed» and «perturbed» differentiation operators on an interval

The article considers the eigenvalue problem of the differentiation operator, when the spectral param- eter is also present in the boundary condition with an integral perturbation, where the integrand has the property of limited variation and has a value of unity at the ends of the segment [-1, 1]. The derivatives with respect to the time variable included in the boundary condition naturally arise when solving (by the Fourier method) initial boundary value problems for evolution equations. The conjugate operator is constructed. It is shown that the spectral questions of the conjugate operator have a similar structure. The characteristic determinant of the original direct spectral problem with an integral perturbation of the boundary condition and for the eigenvalue problem of a loaded first-order differential equation on a segment with a periodic boundary condition, which is an entire analytical function of the spectral parameter, is constructed. Based on the formula of the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the original «perturbed» differentiation operator and the loaded first-order differential equation on the segment. A special feature of the operator under consideration is the non-self-adjointness of the operator in L2(–1,1). The quadratic proximity of the systems of eigen functions of the «unperturbed» and «perturbed» differentiation operators and the Riesz basis property of these systems are proved. In this case, the system of eigenfunctions of the «perturbed» operator is not orthonormal.

Parallel finite-element codes for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates

We present and distribute a parallel finite-element toolbox written in the free software FreeFEM for computing the Bogoliubov-de Gennes (BdG) spectrum of stationary solutions to one- and two-component Gross-Pitaevskii (GP) equations, in two or three spatial dimensions. The parallelization of the toolbox relies exclusively upon the recent interfacing of FreeFEM with the PETSc library. The latter contains itself a wide palette of state-of-the-art linear algebra libraries, graph partitioners, mesh generation and domain decomposition tools, as well as a suite of eigenvalue solvers that are embodied in the SLEPc library. Within the present toolbox, stationary states of the GP equations are computed by a Newton method. Branches of solutions are constructed using an adaptive step-size continuation algorithm. The combination of mesh adaptivity tools from FreeFEM with the parallelization features from PETSc makes the toolbox efficient and reliable for the computation of stationary states. Their BdG spectrum is computed using the SLEPc eigenvalue solver. We perform extensive tests and validate our programs by comparing the toolbox's results with known theoretical and numerical findings that have been reported in the literature. Program summaryProgram Title: FFEM_BdG_ddm_toolbox.zipCPC Library link to program files:https://doi.org/10.17632/w9hg964wpb.1Licensing provisions: GPLv3Programming language:FreeFEM (v 4.12) free software (www.freefem.org)Nature of problem: Among the plethora of configurations that may exist in Gross-Pitaevskii (GP) equations modeling one or two-component Bose-Einstein condensates, only the ones that are deemed spectrally stable (or even, in some cases, weakly unstable) have high probability to be observed in realistic ultracold atoms experiments. To investigate the spectral stability of solutions requires the numerical study of the linearization of GP equations, the latter commonly known as the Bogoliubov-de Gennes (BdG) spectral problem. The present software offers an efficient and reliable tool for the computation of eigenvalues (or modes) of the BdG problem for a given two- or three-dimensional GP configuration. Then, the spectral stability (or instability) can be inferred from its spectrum, thus predicting (or not) its observability in experiments.Solution method: The present toolbox in FreeFEM consists of the following steps. At first, the GP equations in two (2D) and three (3D) spatial dimensions are discretized by using P2 (piece-wise quadratic) Galerkin triangular (in 2D) or tetrahedral (in 3D) finite elements. For a given configuration of interest, mesh adaptivity in FreeFEM is deployed in order to reduce the size of the problem, thus reducing the toolbox's execution time. Then, stationary states of the GP equations are obtained by a Newton method whose backbone involves the use of a reliable and efficient linear solver judiciously selected from the PETSc1 library. Upon identifying stationary configurations, to trace branches of such solutions a parameter continuation method over the chemical potential in the GP equations (effectively controlling the number of atoms in a BEC) is employed with step-size adaptivity of the continuation parameter. Finally, the computation of the stability of branches of solutions (i.e. the BdG spectrum), is carried out by accurately solving, at each point in the parameter space, the underlying eigenvalue problem by using the SLEPc2 library. Three-dimensional computations are made affordable in the present toolbox by using the domain decomposition method (DDM). In the course of the computation, the toolbox stores not only the solutions but also the eigenvalues and respective eigenvectors emanating from the solution to the BdG problem. We offer examples for computing stationary configurations and their BdG spectrum in one- and two-component GP equations.Running time: From minutes to hours depending on the mesh resolution and space dimension.

The novel triangular spectral indices for characterizing winter wheat drought

Agricultural drought threatens food security and agricultural sustainable development. There have been numerous spectral indices from remote sensing images developed for monitoring crop drought. However, most present spectral indices are focusing on crop growth and Land Surface Temperature (LST), and the crop canopy water content are in less consideration simultaneously. Additionally, the Normalized Difference Vegetation Index (NDVI) is used for characterizing crop growth in almost all spectral drought indices, with the spectral saturation problem of NDVI for closed crop canopy. When vegetation cover is high, NDVI values tend to saturate, which makes them insensitive to further changes in crop health. Therefore, the NDVI saturation phenomenon may lead to an underestimation of the extent of crop drought, as it is not effective in identifying subtle changes in crops under high-density vegetation conditions. Hence, we propose three novel triangular spectral indices for characterizing winter wheat drought using three features including LAI, Land Surface Water Index (LSWI) and LST. For validating the proposed spectral indices, we compared the agreement between these indices with measured Relative Soil Moisture (RSM) and Volumetric Water Content (VWC) of soil in agricultural meteorological station and present popular indices including Crop Water Stress Index (CWSI), Temperature-Vegetation Drought Index (TVDI), and Vegetation Health Index (VHI). The results revealed that our proposed indices including Euclidean distance Crop Health Index (ECHI), Difference Crop Health Index (DCHI) and Perpendicular Water Stress Index (PWSI) outperformed the popular CWSI, TVDI and VHI, with stronger correlations with measured RSM and VWC in agricultural meteorological station. Secondly, there are spatial consistencies for characterizing winter wheat drought between proposed ECHI, DCHI and PWSI with popular CWSI, TVDI and VHI. In addition, our proposed ECHI, DCHI and PWSI have achieved good performance of drought monitoring both in irrigated and rainfed croplands. All these results suggest that our proposed indices have great potential in crop drought monitoring.

Generalized Multiscale Finite Element Method for discrete network (graph) models

In this paper, we consider a time-dependent discrete network model with highly varying connectivity. The approximation by time is performed using an implicit scheme. We propose the coarse scale approximation construction of network models based on the Generalized Multiscale Finite Element Method. An accurate coarse-scale approximation is generated by solving local spectral problems in sub-networks. Convergence analysis of the proposed method is presented for semi-discrete and discrete network models. We establish the stability of the multiscale discrete network. Numerical results are presented for structured and random heterogeneous networks.

Multi-geometric discrete spectral problem with several pairs of zeros for Sasa–Satsuma equation

Sasa–Satsuma equation is proposed to model the propagation and interaction of the sub-picosecond or femtosecond pulses in a monomode optical fiber. Different from several integrable equations in the Ablowitz–Kaup–Newell–Segur system, the higher-order zeros of Riemann–Hilbert problem for the Sasa–Satsuma appear in quadruples. A new approach to study the multi-geometric discrete spectral problem with several pairs of zeros for the Sasa–Satsuma equation is proposed. Thus, the complete soliton solutions corresponding to the higher-order zeros with arbitrary geometric and algebraic multiplicities are derived. Moreover, the inelastic interactions between or among the solitons corresponding to the higher-order non-elementary zeros exhibit the shape-changing phenomena.