Spectral Problem Research Articles

The localized excitation on the Weierstrass elliptic function periodic background for the (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili equation

In this paper, the linear spectral problem associated with the (3+1)-dimensional generalized variable-coefficient Kadomtsev-Petviashvili (gvcKP) equation with the Weierstrass function as the external potential is investigated based on the Lamé function, from which some new localized nonlinear wave solutions on the Weierstrass elliptic ℘-function periodic background are obtained by the Darboux transformation. The degenerate solutions on the ℘-function periodic background for the gvcKP equation can be derived by taking the limits of the half-periods ω 1, ω 2 of ℘(x), whose evolution and corresponding dynamics are also discussed. The findings show that nonlinear waves on the ℘-function periodic background behave as different types of nonlinear waves in different spaces, including periodic waves, vortex solitons and interaction solutions, aiding in elucidating some physical phenomena in the related fields, such as the physical ocean and nonlinear optics.

Algebraic solitons in the massive Thirring model.

We present exact solutions describing dynamics of two algebraic solitons in the massive Thirring model. Each algebraic soliton corresponds to a simple embedded eigenvalue in the Kaup-Newell spectral problem and attains the maximal mass among the family of solitary waves traveling with the same speed. By coalescence of speeds of the two algebraic solitons, we find a new solution for an algebraic double-soliton which corresponds to a double embedded eigenvalue. We show that the double-soliton attains the double mass of a single soliton and describes a slow interaction of two identical algebraic solitons.

Finite element analysis for the Navier-Lamé eigenvalue problem

The present paper introduces the analysis of the eigenvalue problem for the elasticity equations when the so-called Navier-Lamé system is considered. This system incorporates the displacement, rotation, and pressure of a linear elastic structure. The analysis of the spectral problem is based on the compact operator theory. A finite element method using polynomials of degree k≥1 is employed to approximate the eigenfrequencies and eigenfunctions of the system. Convergence and error estimates are presented. An a posteriori error analysis is also performed, where the reliability and efficiency of the proposed estimator are proven. We conclude this contribution by reporting a series of numerical tests to assess the performance of the proposed numerical method for both a priori and a posteriori estimates.

Solitons with varying amplitudes and/or varying velocities in a variable-coefficient mixed spectral generalized Sasa–Satsuma equation revealed through the Riemann–Hilbert method

The integrable Sasa–Satsuma (SS) equation, a higher-order Nonlinear Schrödinger (NLS)-type model, has important applications in the field of optics. In this study, we present a generalized SS equation with variable coefficients derived from a mixed spectral problem, abbreviated as vcmsgSSE, and construct its N-soliton solution using the Riemann–Hilbert (RH) method. First, we provide the Lax pair associated with the vcmsgSSE and perform a spectral analysis on it, which allows us to derive a solvable RH problem. Then, by solving this RH problem we construct an explicit expression for N-soliton solution of the vcmsgSSE. Finally, under a special reduction, we obtain the specific one-soliton solution and two-soliton solution of the vcmsgSSE, and use them to illustrate graphically the N-soliton solution with varying amplitude and/or varying velocity corresponds to the physical background of non-uniform medium assumed by the vcmsgSSE.

Scaling law for a buckled elastic filament in a shear flow.

We analyze the three-dimensional (3D) buckling of an elastic filament in a shear flow of a viscous fluid at low Reynolds number and high Péclet number. We apply the Euler-Bernoulli beam (elastica) theoretical model. We show the universal character of the full 3D spectral problem for a small perturbation of a thin filament from a straight position of arbitrary orientation. We use the eigenvalues and eigenfunctions for the linearized elastica equationin the shear plane, found earlier by Liu etal. [Phys. Rev. Fluids 9, 014101 (2024)2469-990X10.1103/PhysRevFluids.9.014101] with the Chebyshev spectral collocation method, to solve the full 3D eigenproblem. We provide a simple analytic approximation of the eigenfunctions, represented as Gaussian wave packets. As the main result of the paper, we derive the square-root dependence of the eigenfunction wave number on the parameter χ[over ̃]=-ηsin2ϕsin^{2}θ, where η is the elastoviscous number and the filament orientation is determined by the zenith angle θ with respect to the vorticity direction and the azimuthal angle ϕ relative to the flow direction. We also compare the eigenfunctions with shapes of slightly buckled elastic filaments with a non-negligible thickness with the same Young's modulus, using the bead model and performing numerical simulations with the precise hydromultipole numerical codes.

Optical solitons and their propagations in a time-varying coefficient modified nonlinear Schrödinger equation with non-zero boundary conditions via Riemann–Hilbert method

In this article, a time-varying coefficient modified nonlinear Schrödinger equation (tvcmNLSE) with non-zero boundary conditions (NZBCs) at infinity, phase term of which contains time-varying coefficients, is proposed to describe the nonlinear propagation behavior of optical pulses in non-uniform optical fibre. Meanwhile, the Riemann–Hilbert method (RHM) for the tvcmNLSE with NZBCs is presented for the first time. Specifically, a special transformation is constructed to convert the tvcmNLSE with NZBCs into the case with constant NZBCs, and the associated asymptotic spectral problem (ASP) is obtained. By introducing a suitable two-piece Riemann surface to map the original spectral parameter into a single-valued one, and analyzing the ASP, we construct the corresponding Jost solutions and spectral matrix, and sequentially analyze their analytical, symmetric, and asymptotic properties. Then, a generalized Riemann–Hilbert problem (RHP) is established that relates to the converted tvcmNLSE with constant NZBCs. Using the reconstruction formula, we further derive N-soliton solution with NZBCs in the absence of reflection potential, and analyze the dynamic characteristics of single and double solitons. Most notably, by modulating discrete spectral points (DSPs) and time-varying coefficients, we discover some previously unreported new waveforms, such as parabolic and periodic solitons with NZBCs. These waveforms have important implications in both theory and practice. In addition, the influences of different time-varying coefficients, NZBCs, and DSPs on soliton solutions are also discussed and analyzed. This study shows that the selection of time-varying coefficients, distribution of DSPs, and constraints from NZBCs all have significant impacts on the properties of soliton solutions.

RANK OF THE DERIVATIVE OF THE PROJECTION TO SYMMETRIZED POLYDISC

The projection, also called the symmetrization mapping, from spectral ball to symmetrized polydisc is closely related to the spectral Nevanlinna-Pick interpolation problem. We prove that the rank of the derivative of the projection from the spectral unit ball to the symmetrized polydisc is equal to the degree of the minimal polynomial of the matrix at which we take the derivative. Therefore, it explains why the corresponding lifting problem is easier when the matrix base-point is cyclic since it is a regular point of the symmetrization mapping in the differential sense.

On spectral eigenmatrix problem for the planar self-affine measures with three digits

On spectral eigenmatrix problem for the planar self-affine measures with three digits

MKdV Equation on Time Scales: Darboux Transformation and N-Soliton Solutions

In this paper, the spectral problem of the mKdV equation satisfying the compatibility condition on time scales is directly constructed. By using the zero-curvature equation on time scales, the mKdV equation on time scales is obtained. When x∈R and t∈R, the equation degenerates to the classical mKdV equation. Then, the single-soliton, two-soliton, and N-soliton solutions of the mKdV equation under the zero boundary condition on time scales are presented via employing the Darboux transformation (DT). Particularly, we obtain the corresponding single-soliton solutions expressed using the Cayley exponential function on four different time scales (R, Z, q-discrete, C).

Disentanglement-inspired single-source domain-generalization network for cross-scene hyperspectral image classification

Cross-scene classification stands as a pivotal frontier in hyperspectral image (HSI) processing, aiming to enhance the generalization capabilities of classification models. However, the diversity of sensor type, shooting environments, and shooting times leads to the spectral heterogeneity problem in HSI. As a result, the same land cover may exhibit varying spectral traits in different domains, posing challenges for cross-scene HSI classification. Drawing inspiration from image disentanglement, we have identified that extracting the latent domain-invariant representation (DIR) of HSI could potentially mitigate the spectral heterogeneity issue. Therefore, we propose a Disentanglement-Inspired Single-Source Domain Generalization Network (DSDGnet) for cross-scene HSI classification in this paper. Firstly, a style transfer module based on a Transformer encoder-transfer-decoder is designed to expand the single source domain to an extended domain. Then, a progressive disentanglement module is proposed to decompose the domain-invariant features and domain-specific features of HSI. Furthermore, a domain combination module is designed to guarantee the accuracy of the progressive disentanglement module and ensure the effectiveness of the domain-invariant feature of HSI. Finally, the domain-invariant features are applied to the classification task, and the domain-specific features are separated to reduce their impact on the generalization ability of classification models. Extensive experiments on three HSI datasets have demonstrated the advanced classification performance of DSDGnet compared to existing domain-generalization methods.

Continuity and minimization of spectrum related with the two-component Novikov equation

This article is concerned with the spectral problem{dϕ1(x)=ϕ2(x)dh(x),dϕ2(x)=ϕ3(x)dg(x),dϕ3(x)=−λϕ1(x)dx, x∈(−1,1),ϕ2(−1)=0, ϕ3(−1)=0, ϕ3(1)=0,which is associated with the two-component Novikov equation. The coefficients g and h are functions of bounded variation. The first aim is to prove the continuous dependence of the n-th eigenvalue on the coefficients g and h with different topologies; in addition, we discuss the Fréchet differentiability of the algebraically simple eigenvalues on the coefficients g, h with respect to the norm topology of total variations. Secondly, utilizing the theory of oscillatory (non-symmetric) kernels, conditions on g, h under which the spectrum is nonnegative and algebraically simple are obtained. Finally, we solve the extremal problems of the lowest nonzero eigenvalue, when the norm ‖⋅‖V of the coefficients g, h is given. The latter can be seen as an application of our first two main results and the trace formulas.

Bi-Hamiltonian structure of a super KdV equation of Kupershmidt

In this paper, we study a super Korteweg–de Vries (sKdV) equation proposed by Kupershmidt which possesses a Lax operator with three fully nonlocal terms. The Lax operator is reformulated so that it is of the super constrained modified Kadomtsev–Petviashvili (scmKP) type. By calculating the bi-Hamiltonian structure of the scmKP hierarchy and employing Dirac reduction, we obtain the bi-Hamiltonian structure of the sKdV equation. We also present a spectral problem of its modified system.

Coherent distributions of the symmetry algebra of spinor regularization generator and hydrogen atom

A new approach is developed for solving spectral problems for operators with continuous spectrum, which consists in the integral transform of the problem by using coherent Schwartz distributions. The constructed family of coherent distributions is a complete analogue of the family of ordinary coherent states. More precisely, it satisfies all Gazeau–Klauder axioms satisfied by the usual coherent states. But in contrast to the coherent states belonging to the point spectrum of the annihilation operator (or operators), the coherent distributions belong to the continuous spectrum of some Hermitian operators. Therefore, the coherent distributions work better than the coherent states as the kernel of the integral representation of generalized eigenfunctions of operators with continuous spectrum. In this work, this approach is demonstrated with an example of solving a basic problem of quantum mechanics, i.e., the problem of the continuous part of the spectrum of the Hamiltonian of the hydrogen atom.

Solving the spectral problem via the boundary approximation in theory

In theory, the resonance scattering structure is triggered by the so-called delocalized modes trapped between the pair. The frequencies and configurations of such modes depend on the half-separation a, which can be derived from the Schrödinger-like equation. We propose to use the boundary conditions to connect the half-localized and delocalized modes, and use boundary approximation (BA) to solve the spectrum analytically. In detail, we derive the explicit form of frequencies, configurations and spectral wall locations of the delocalized modes. We test the analytical prediction with the numerical simulation of the Schrödinger-like equation, and obtain astonishing agreement between them at the long separation regime.

On the positive invertibility of a differential operator on a graph

In this article, we study a fourth-order differential operator L λ on a graph, depending on a real parameter λ. The operator L λ is connected with a model of the Euler–Bernoulli beam system. Our consideration is devoted to the study of the set of values of the spectral parameter λ for which the operator L λ is inverse positive. It is shown that the operator L λ is positively invertible if and only if there exists a fundamental system of solutions to the corresponding homogeneous equation, consisting of functions that are positive on the graph. We establish necessary and sufficient conditions for the differential operator L λ to be inverse-positive for all positive values of the spectral parameter less than the lowest eigenvalue of the differential operator L 0 corresponding to λ = 0 . In this way, we establish the positivity of eigenvalues and formulate comparison theorems for eigenvalues of the spectral problem. Next, we formulate maximum principles for a fourth-order differential inequalities on a graph.

A discontinuous Galerkin based multiscale method for heterogeneous elastic wave equations

In this paper, we develop a local multiscale model reduction strategy for the elastic wave equation in strongly heterogeneous media, which is achieved by solving the problem in a coarse mesh with multiscale basis functions. We use the interior penalty discontinuous Galerkin (IPDG) to couple the multiscale basis functions that contain important heterogeneous media information. The construction of efficient multiscale basis functions starts with extracting dominant modes of carefully defined spectral problems to represent important media feature, which is followed by solving a constraint energy minimization problem. Then a Petrov-Galerkin projection and systematization onto the coarse grid are applied. As a result, an explicit and energy-conserving scheme is obtained for fast online simulation. The method exhibits both coarse-mesh and spectral convergence as long as one appropriately chooses the oversampling size. We rigorously analyze the stability and convergence of the proposed method. Numerical results are provided to show the performance of the multiscale method and confirm the theoretical results.

Approximation of eigenvalues and eigenfunctions of the diffusion operator in a domain containing thin tubes by asymptotic domain decomposition method

We consider the spectral problem for the diffusion operator considered in a domain containing thin tubes. A new version of the method of partial asymptotic decomposition of the domain is introduced to reduce the dimension inside the tubes, getting a model of hybrid dimensions. The method truncates the tubes at some small distance from the ends of the tubes and replaces the longer part of the tubes with segments. At the interface of the three-dimensional and one-dimensional subdomains, special junction conditions are set: the pointwise continuity of the flux and the continuity of the average over a cross-section of the eigenfunctions. We obtain conditions on the ratio of the characteristic sizes in the transverse and longitudinal directions that ensure the closeness of two spectra, i.e. of the diffusion operator in the full-dimensional domain and the partially reduced one, keeping the conservation of the multiplicity, all up to a prescribed accuracy.

Existence of Eigenvalues and Boundedness of the Eigenfunctions and the Orthogonality of the Eigenfunctions of a Type of Spectral Problem

Background: This research aims to investigate certain requirements for the presence of eigenvalues, as well as the boundaries of eigenfunctions and their derivatives, specifically, the eigenfunctions' first as well as second derivatives. Materials and Methods: In this study, we use the spectral problem of second-order differential equations: , with mixed boundary conditions , where is a spectral parameter. and the normalized condition , where . Results: We get that the spectral parameter of a second-order differential operator is real. And we obtain Lagrange’s identity for a spectral problem. Also, we prove that the spectral problem is self-adjoint, and the property of orthogonality of eigenfunctions is shown. Conclusions: In this research, we studied the existence of eigenvalues and the estimation of the norm of eigenfunctions for problem (1) - (3). Furthermore, we investigated the self-adjoint nature of the problem, and we proved that the eigenfunctions are orthogonal.

On the spectral problem for three-dimesional bi-Laplacian in the unit sphere

On the spectral problem for three-dimesional bi-Laplacian in the unit sphere

Inverse problems for the eigenparameter Dirac operator with complex weight

Abstract Inverse spectral problems are considered for the discontinuous Dirac operator with complex-value weight and the spectral parameter boundary conditions. We investigate some properties of spectral characteristics and show that the potential can be uniquely determined by the Weyl-type function or by two spectra on the whole interval.