Spectral Problem Research Articles

Inverse problems for Dirac operators with constant delay less than half of the interval

In this work, we consider Dirac-type operators with a constant delay of less than half of the interval and not less than two-fifths of the interval. For our considered Dirac-type operators, two inverse spectral problems are studied. Specifically, reconstruction of two complex L2-potentials is studied from complete spectra of two boundary value problems with one common boundary condition y1(0) = 0 or y2(0) = 0. We give answers to the full range of questions usually raised in the inverse spectral theory. That is, we give uniqueness, necessary and sufficient conditions of the solvability, reconstruction algorithm and uniform stability for our considered inverse problems.

The τ$$ \tau $$‐symmetries and Lie algebra structure of the Blaszak–Marciniak lattice equation

A general approach is developed for discriminating strong and hereditary symmetric operators. The recursion operator of the Blaszak–Marciniak (BM) equation hierarchy is proved to be strong and hereditary symmetric. As an example of discrete soliton equations related to matrix spectral problems, the ‐symmetries and Lie algebra structure of the BM equation are built firstly.

Construction of homogeneous solutions of the torsion problem for a radially inhomogeneous transversely isotropic cylinder

The torsion problem for a radially inhomogeneous transversely isotropic cylinder of small thickness was investigated by the method of asymptotic integration of elasticity theory equations. It is assumed that the side part of the cylinder is stress-free, and boundary conditions are set at the ends of the cylinder, leaving the cylinder in equilibrium. The elastic moduli are thought to be arbitrary continuous functions of the variable along the cylinder radius. The formulated boundary value problem is reduced to a spectral problem containing a small parameter characterizing the thin-walledness of the cylinder. Homogeneous solutions are built, i.e. any solutions of the equilibrium equation satisfying the condition of no stresses on the side surfaces. It is shown that the solution of the torsion problem consists of a penetrating solution and a boundary layer character solution similar to Saint-Venant's edge effect in the theory of inhomogeneous plates. 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 torsional moments of stresses acting in the cross-section perpendicular to the cylinder axis. Solutions having the boundary layer character are localized at the ends of the cylinder and decrease exponentially with distance from the ends. These solutions are absent in applied shell theories. Asymptotic formulas for displacement and stresses are built, which make it possible to calculate the three-dimensional stress-strain state of a radially inhomogeneous transversely isotropic cylinder of small thickness. Based on the obtained asymptotic expansions, it is possible to assess the applicability of applied theories and build a refined applied theory for radially inhomogeneous cylindrical shells

Construction of exact and asymptotic solutions for the problem of torsional vibrations of a radially inhomogeneous sphere

In this paper, the torsional vibrations of a radially inhomogeneous isotropic open sphere containing no 0 and π poles are studied. It is considered that the elastic moduli and density of the material are linear functions of the radius of the sphere. Cases are considered when the lateral surface of the sphere is stress-free and fixed, and arbitrary boundary conditions are specified on conic sections. The formulated boundary value problems are reduced to spectral problems. After fulfilling boundary conditions specified on the lateral surfaces of the sphere, a dispersion equation is obtained. Exact solutions to the problem are constructed. Next, we analyzed the roots of the dispersion equation for a relatively small parameter characterizing the thickness of the sphere, depending on the oscillation frequency. For a sphere of small thickness, asymptotic zeros are constructed, and their characteristics are studied.

A generalized Hermite–Biehler theorem and non-Hermitian perturbations of Jacobi matrices

The classical Hermite–Biehler theorem describes the zero configuration of a complex linear combination of two real polynomials whose zeros are real, simple, and strictly interlace. We provide the full characterization of the zero configuration for the case when this interlacing is broken at exactly one location. We apply this result to solve the direct and inverse spectral problem for non-Hermitian rank-one multiplicative perturbations and rank-two additive perturbations of finite Hermitian and Jacobi matrices.

Rotation number and eigenvalues of two-component modified Camassa–Holm equations

In this paper, we study the spectral problem y1y2x=−1212λm−12λn12y1y2for the two-component modified Camassa–Holm equation, where m,n are two potentials. By introducing the rotation number ρ(λ) and studying its properties, we prove that for any integer k, the periodic or anti-periodic eigenvalues are the endpoints of the interval λ∈R:ρ(λ)=−k2. Moreover, we prove that as nonlinear functionals of potentials, such eigenvalues are continuous in potentials with respect to the weak topologies in the Lebesgue space L1[0,T]. Finally, we apply the trace formula to give some estimates of the periodic eigenvalues when the L1 norms of potentials are given.

Spectral problem for the complex mKdV equation: singular manifold method and Lie symmetries

This article addresses the study of the complex version of the modified Korteweg-de Vries equation using two different approaches. Firstly, the singular manifold method is applied in order to obtain the associated spectral problem, binary Darboux transformations and $\tau$-functions. The second part concerns the identification of the classical Lie symmetries for the spectral problem. The similarity reductions associated to these symmetries allow us to derive the reduced spectral problems and first integrals for the ordinary differential equations arising from such reductions.

On the Links between Miura Transformations of Bogoyavlensky Lattices and Inverse Spectral Problems for Band Operators

We consider semi-infinite and finite Bogoyavlensky lattices$$\overset\cdot a_i =a_i\left(\prod_{j=1}^{p}a_{i+j}-\prod_{j=1}^{p}a_{i-j}\right),$$$$\overset\cdot b_i = b_i\left(\sum_{j=1}^{p}b_{i+j}-\sum_{j=1}^{p}b_{i-j}\right),$$for some $p\ge 1$, and Miura-like transformations between these systems, defined for $p\ge 2$. Both lattices are integrable (via Lax pair formalism) by the inverse spectral problem method for band operators, i.e., operators generated by band matrices. The key role in this method is played by the moments of the Weyl matrix of the corresponding band operator and their evolution in time. We find a description of the above-mentioned transformations in terms of these moments and apply this result to study finite Bogoyavlensky lattices and, in particular, their first integrals.

Extending third quantization with commuting observables: a dissipative spin-boson model

We consider the spectral and initial value problem for the Lindblad–Gorini–Kossakowski–Sudarshan master equation describing an open quantum system of bosons and spins, where the bosonic parts of the Hamiltonian and Lindblad jump operators are quadratic and linear respectively, while the spins couple to bosons via mutually commuting spin operators. Needless to say, the spin degrees of freedom can be replaced by any set of finite-level quantum systems. A simple, yet non-trivial example of a single open spin-boson model is worked out in some detail.

Breather wave solutions on the Weierstrass elliptic periodic background for the (2 + 1)-dimensional generalized variable-coefficient KdV equation.

In this paper, the Nth Darboux transformations for the (2+1)-dimensional generalized variable-coefficient Koretweg-de Vries (gvcKdV) equation are proposed. By using the Lamé function method, the generalized Lamé-type solutions for the linear spectral problem associated with the gvcKdV equation with the static and traveling Weierstrass elliptic ℘-function potentials are derived, respectively. Then, the nonlinear wave solutions for the gvcKdV equation on the static and traveling Weierstrass elliptic ℘-function periodic backgrounds under some constraint conditions are obtained, respectively, whose evolutions and dynamical properties are also discussed. The results show that the degenerate solutions on the periodic background can be obtained by taking the limits of the half-periods ω1,ω2 of ℘(x), and the evolution curves of nonlinear wave solutions on the periodic background are determined by the coefficients of the gvcKdV equations.

A second-order exponential integration constraint energy minimizing generalized multiscale method for parabolic problems

This paper investigates an efficient exponential integrator generalized multiscale finite element method for solving a class of time-evolving partial differential equations in bounded domains. The proposed method first performs the spatial discretization of the model problem using constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM). This approach consists of two stages. First, the auxiliary space is constructed by solving local spectral problems, where the basis functions corresponding to small eigenvalues are captured. The multiscale basis functions are obtained in the second stage using the auxiliary space by solving local energy minimization problems over the oversampling domains. The basis functions have exponential decay outside the corresponding local oversampling regions. We shall consider the first and second-order explicit exponential Runge-Kutta approach for temporal discretization and to build a fully discrete numerical solution. The exponential integration strategy for the time variable allows us to take full advantage of the CEM-GMsFEM as it enables larger time steps due to its stability properties. We derive the error estimates in the energy norm under the regularity assumption. Finally, we will provide some numerical experiments to sustain the efficiency of the proposed method.

Automatic detection system with efficient and accurate sample preparation for cobalt ion concentration in zinc hydrometallurgy

Cobalt ion is an impurity that has a negative effect in zinc hydrometallurgy, resulting in lower zinc yield and quality. Currently, the zinc hydrometallurgy industry typically uses manual sampling followed by manual sample preparation, and single-wavelength spectral analysis to detect cobalt ions in zinc solution, which is inefficient and lacks accuracy. Therefore, a new automatic detection system for cobalt ions concentration is developed. The system effectively overcomes challenges of pipeline blockage and corrosion by innovatively implementing negative pressure extraction, volume quantification, and split pipeline structure. Compared with manual detection, the system has higher detection repeatability and efficiency, specifically reducing reagent consumption by 70 %, and improving the detection speed by 8 times. In addition, to solve the spectral drift and noise problems, a spectral preprocessing algorithm by forced contraction of long-wavelength region spectra and Gaussian filtering is proposed. This algorithm not only improves the local linearity and regression accuracy of the spectrum, but also improves the detection repeatability of the system. Finally, five full-wavelength modeling methods are applied to the system. The best root mean square error for low concentration (0.1 to 1 mg/L) and high concentration (1 to 20 mg/L) cobalt zinc solution is only 0.0027 mg/L and 0.071 mg/L, respectively.

Uniqueness of the extremal Schwarzschild de Sitter spacetime

We prove that any analytic vacuum spacetime with a positive cosmological constant in four and higher dimensions, that contains a static extremal Killing horizon with a maximally symmetric compact cross-section, must be locally isometric to either the extremal Schwarzschild de Sitter solution or its near-horizon geometry (the Nariai solution). In four-dimensions, this implies these solutions are the only analytic vacuum spacetimes that contain a static extremal horizon with compact cross-sections (up to identifications). We also consider the analogous uniqueness problem for the four-dimensional extremal hyperbolic Schwarzschild anti-de Sitter solution and show that it reduces to a spectral problem for the laplacian on compact hyperbolic surfaces, if a cohomological obstruction to the uniqueness of infinitesimal transverse deformations of the horizon is absent.

A ∂¯-Dressing Method for the Kundu-Nonlinear Schrödinger Equation

In this paper, we employed the ∂¯-dressing method to investigate the Kundu-nonlinear Schrödinger equation based on the local 2 × 2 matrix ∂¯ problem. The Lax spectrum problem is used to derive a singular spectral problem of time and space associated with a Kundu-NLS equation. The N-solitions of the Kundu-NLS equation were obtained based on the ∂¯ equation by choosing a special spectral transformation matrix, and a gradual analysis of the long-duration behavior of the equation was acquired. Subsequently, the one- and two-soliton solutions of Kundu-NLS equations were obtained explicitly. In optical fiber, due to the wide application of telecommunication and flow control routing systems, people are very interested in the propagation of femtosecond optical pulses, and a high-order, nonlinear Schrödinger equation is needed to build a model. In plasma physics, the soliton equation can predict the modulation instability of light waves in different media.

Practical Aspects of Physics-Informed Neural Networks Applied to Solve Frequency-Domain Acoustic Wave Forward Problem

Abstract Physics-informed neural networks (PINNs) have been used by researchers to solve partial differential equation (PDE)-constrained problems. We evaluate PINNs to solve for frequency-domain acoustic wavefields. PINNs can solely use PDEs to define the loss function for optimization without the need for labels. Partial derivatives of PDEs are calculated by mesh-free automatic differentiations. Thus, PINNs are free of numerical dispersion artifacts. It has been applied to the scattered acoustic wave equation, which relied on boundary conditions (BCs) provided by the background analytical wavefield. For a more direct implementation, we solve the nonscattered acoustic wave equation, avoiding limitations related to relying on the background homogeneous medium for BCs. Experiments support our following insights. Although solving time-domain wave equations using PINNs does not require absorbing boundary conditions (ABCs), ABCs are required to ensure a unique solution for PINNs that solve frequency-domain wave equations, because the single-frequency wavefield is not localized and contains wavefield information over the full domain. However, it is not trivial to include the ABC in the PINN implementation, so we develop an adaptive amplitude-scaled and phase-shifted sine activation function, which performs better than the previous implementations. Because there are only two outputs for the fully connected neural network (FCNN), we validate a linearly shrinking FCNN that can achieve a comparable and even better accuracy with a cheaper computational cost. However, there is a spectral bias problem, that is, PINNs learn low-frequency wavefields far more easily than higher frequencies, and the accuracy of higher frequency wavefields is often poor. Because the shapes of multifrequency wavefields are similar, we initialize the FCNN for higher frequency wavefields by that of the lower frequencies, partly mitigating the spectral bias problem. We further incorporate multiscale positional encoding to alleviate the spectral bias problem. We share our codes, data, and results via a public repository.

Sharp-Interface Limit of a Multi-phase Spectral Shape Optimization Problem for Elastic Structures

We consider an optimization problem for the eigenvalues of a multi-material elastic structure that was previously introduced by Garcke et al. (Adv. Nonlinear Anal. 11:159–197, 2022). There, the elastic structure is represented by a vector-valued phase-field variable, and a corresponding optimality system consisting of a state equation and a gradient inequality was derived. In the present paper, we pass to the sharp-interface limit in this optimality system by the technique of formally matched asymptotics. Therefore, we derive suitable Lagrange multipliers to formulate the gradient inequality as a pointwise equality. Afterwards, we introduce inner and outer expansions, relate them by suitable matching conditions and formally pass to the sharp-interface limit by comparing the leading order terms in the state equation and in the gradient equality. Furthermore, the relation between these formally derived first-order conditions and results of Allaire and Jouve (Comput. Methods Appl. Mech. Eng. 194:3269–3290, 2005) obtained in the framework of classical shape calculus is discussed. Eventually, we provide numerical simulations for a variety of examples. In particular, we illustrate the sharp-interface limit and also consider a joint optimization problem of simultaneous compliance and eigenvalue optimization.

An extended Merola–Ragnisco–Tu lattice integrable hierarchy and infinite conservation laws

By means of the zero-curvature equation and Lenard recursive operators, a novel differential-difference integrable hierarchy is derived, which is related to a discrete 3 × 3 matrix spectral problem with five potentials. The first nontrivial member in this hierarchy under specific reduction is the Merola–Ragnisco–Tu lattice equation whose continuum limit is the nonlinear Schrödinger equation. The infinite conservation laws of the first two nontrivial members in the hierarchy, namely, the extended Merola–Ragnisco–Tu lattice equation and its high-order generalization are constructed with the help of two Ricatti-type equations and the spectral parameter expansions.

Stable Singularity Formation for the Keller–Segel System in Three Dimensions

We consider the parabolic–elliptic Keller–Segel system in dimensions d≧3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d \\geqq 3$$\\end{document}, which is the mass supercritical case. This system is known to exhibit rich dynamical behavior including singularity formation via self-similar solutions. An explicit example was found more than two decades ago by Brenner et al. (Nonlinearity 12(4):1071–1098, 1999), and is conjectured to be nonlinearly radially stable. We prove this conjecture for d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document}. Our approach consists of reformulating the problem in similarity variables and studying the Cauchy evolution in intersection Sobolev spaces via semigroup theory methods. To solve the underlying spectral problem, we use a technique we recently established in Glogić and Schörkhuber (Comm Part Differ Equ 45(8):887–912, 2020). To the best of our knowledge, this provides the first result on stable self-similar blowup for the Keller–Segel system. Furthermore, the extension of our result to any higher dimension is straightforward. We point out that our approach is general and robust, and can therefore be applied to a wide class of parabolic models.

Long-time asymptotics for the coupled complex short-pulse equation with decaying initial data

We characterize the long-time asymptotic behavior of the solution of the initial value problem for the coupled complex short-pulse equation associated with the 4×4 matrix spectral problem. The spectral analysis of the 4×4 matrix spectral problem is very difficult because of the existence of energy-dependent potentials and the WKI type. The method we adopted is a combination of the inverse scattering transform and Deift-Zhou nonlinear steepest descent method. Starting from the Lax pair associated with the coupled complex short-pulse equation, we derive a basic Riemann-Hilbert problem by introducing some appropriate spectral function transformations, and reconstruct the potential parameterized from the solution of the basic Riemann-Hilbert problem via the asymptotic behavior of the spectral variable at k→0. We finally obtain the leading order asymptotic behavior of the solution of the coupled complex short-pulse equation through a series of Deift-Zhou contour deformations.

Recovering the shape of an equilateral quantum tree with the Dirichlet conditions at the pendant vertices

We consider two spectral problems on an equilateral rooted tree with the standard (continuity and Kirchhoff's type) conditions at the interior vertices (except of the root if it is interior) and Dirichlet conditions at the pendant vertices (except of the root if it is pendant). For the first (Neumann) problem we impose the standard conditions (if the root is an interior vertex) or Neumann condition (if the root is a pendant vertex) at the root, while for the second (Dirichlet) problem we impose the Dirichlet condition at the root. We show that for caterpillar trees the spectra of the Neumann problem and of the Dirichlet problem uniquely determine the shape of the tree. Also, we present an example of co-spectral snowflake graphs.