Third-order Operator Research Articles

Hua operators on homogeneous line bundles over non-tube type bounded symmetric domains

Hua operators on homogeneous line bundles over non-tube type bounded symmetric domains

Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds

Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds

Inverse scattering problem for a third-order operator with local potential

Inverse scattering problem for a third-order operator with local potential

BGG Sequences with Weak Regularity and Applications

We investigate some Bernstein–Gelfand–Gelfand complexes consisting of Sobolev spaces on bounded Lipschitz domains in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^{n}$$\\end{document}. In particular, we compute the cohomology of the conformal deformation complex and the conformal Hessian complex in the Sobolev setting. The machinery does not require algebraic injectivity/surjectivity conditions between the input spaces, and allows multiple input complexes. As applications, we establish a conformal Korn inequality in two space dimensions with the Cauchy–Riemann operator and an additional third-order operator with a background in Möbius geometry. We show that the linear Cosserat elasticity model is a Hodge–Laplacian problem of a twisted de Rham complex. From this cohomological perspective, we propose potential generalizations of continuum models with microstructures.

Exploring network reliability by predicting link status based on simplex neural network

Complex networks are graph-based structures with non-trivial topological features that frequently occur in real systems. Link prediction plays an important role in various real-world networks application, such as recommendation systems, protein structure prediction, packet forwarding strategy optimization, etc. The existing link prediction approaches mainly focus on superficial heuristic features, while ignoring high-order structure information. In this paper, we propose a deep-learning based model, named Weisfeiler-Lehman Simplicial Neural Network (WL-SNN), which can learn the high-order simplex information of the network. In particular, we design a third-order Laplace operator to extract the simplicial features and utilize the graph convolutional network to compensate for the possible deficiencies of the model resulting from the single-channel features. Furthermore, we use the Weisfeiler-Lehman algorithm to extract closed subgraphs of the target, which significantly enhances the adaptability of the model to large-scale networks. Experimental results on six real-world networks show that our approach achieves comparable performance in the link prediction task as well as in the stability analysis of the network.

Inverse scattering problem for a third-order operator with non-local potential

Inverse scattering problem for a third-order operator with non-local potential

Induction Motor Broken Rotor Bar Fault Diagnosis Based on Third-Order Energy Operator Demodulated Current Signal

Motor current signature analysis (MCSA) has been widely used for fault diagnosis of induction motors (IMs) because of its non-invasiveness and comparatively low cost. However, the spectral leakage of the fundamental supply frequency can obscure the fault characteristic component when IMs operating at no-load or low-load conditions. In this paper, a broken rotor bar (BRB) fault diagnosis approach based on the third-order energy operator (TOEO) demodulated current signals is presented. The TOEO approach performs effectively in reducing the negative impact of the fundamental supply frequency spectral leakage and enhancing the BRB fault features. Theoretical and experimental results show that the TOEO approach can excellently reveal the fault characteristic frequency, instead of the sideband frequencies. Besides, the results are compared with the Teager-Kaiser energy operator (TKEO). The capability of the presented approach has several advantages over the TKEO, especially when the IMs are fed by an inverter.

The Cauchy problem for 3-evolution equations with data in Gelfand–Shilov spaces

We consider the Cauchy problem for a third-order evolution operator P with (t, x)-depending coefficients and complex-valued lower-order terms. We assume the initial data to be Gevrey regular with an exponential decay at infinity, that is, the data belong to some Gelfand–Shilov space of type {mathscr {S}}. Under suitable assumptions on the decay at infinity of the imaginary parts of the coefficients of P we prove the existence of a solution with the same Gevrey regularity of the data and we describe its behavior for |x| rightarrow infty .

Inverse spectral problem for a third-order differential operator with non-local potential

Spectral problem for a self-adjoint third-order differential operator with non-local potential on a finite interval is studied. Elementary functions that are analogues of sines and cosines for such operators are described. Direct and inverse problems for third-order operators with non-local potential are solved.

Coupled Nonlinear Dynamics in the Three-Mode Integrable System on a Regular Chain

The article suggests the nonlinear lattice system of three dynamical subsystems coupled both in their potential and kinetic parts. Due to its essentially multicomponent structure the system is capable to model nonlinear dynamical excitations on regular quasi-one-dimensional lattices of various physical origins. The system admits a clear Hamiltonian formulation with the standard Poisson structure. The alternative Lagrangian formulation of system’s dynamics is also presented. The set of dynamical equations is integrable in the Lax sense, inasmuch as it possesses a zero-curvature representation. Though the relevant auxiliary linear problem involves a spectral third-order operator, we have managed to develop an appropriate two-fold Darboux–Backlund dressing technique allowing one to generate the nontrivial crop solution embracing all three coupled subsystems in a rather unusual way.

Conformal boundary operators, T-curvatures, and conformal fractional Laplacians of odd order

We construct continuously parametrised families of conformally invariant boundary operators on densities. These may also be viewed as conformally covariant boundary operators on functions and generalise to higher orders the first-order conformal Robin operator and an analogous third-order operator of Chang-Qing. Our families include operators of critical order on odd-dimensional boundaries. Combined with the (conformal Laplacian power) GJMS operators, a suitable selection of the boundary operators yields formally self-adjoint elliptic conformal boundary problems. Working on a conformal manifold with boundary, we show that the operators yield odd-order conformally invariant fractional Laplacian pseudo-differential operators. To do this, we use higher-order conformally invariant Dirichlet-to-Neumann constructions. We also find and construct new curvature quantities associated to our new operator families. These have links to the Branson Q-curvature and include higher-order generalisations of the mean curvature and the T-curvature of Chang-Qing. In the case of the standard conformal hemisphere, the boundary operator construction is particularly simple; the resulting operators provide an elementary construction of families of symmetry breaking intertwinors between the spherical principal series representations of the conformal group of the equator, as studied by Juhl and others. We use our constructions to shed light on some conjectures of Juhl.

Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians

In this work, we introduce a new realization of exactly-solvable time-dependent Hamiltonians based on the solutions of the fourth Painlevé and the Ermakov equations. The latter is achieved by introducing a shape-invariant condition between an unknown quantum invariant and a set of third-order intertwining operators with time-dependent coefficients. New quantum invariants are constructed after adding a deformation term to the well-known quantum invariant of the parametric oscillator. Such a deformation depends explicitly on time through solutions of the Ermakov equation, a property that simultaneously ensures the regularity of the new time-dependent potentials at each time. The fourth Painlevé equation appears after introducing an appropriate reparametrization of the spatial coordinate and the time parameter, where the parameters of the fourth Painlevé equation dictate the spectral information of the quantum invariant. In this form, the eigenfunctions of the third-order ladder operators lead to several sequences of solutions to the Schrödinger equation, which are determined in terms of the solutions of the Riccati equation, Okamoto polynomials, and nonlinear bound states of the derivative nonlinear Schrödinger equation. Remarkably, it is noticed that the solutions in terms of the nonlinear bound states lead to a quantum invariant with equidistant eigenvalues, which contains both an finite-dimensional and an infinite-dimensional sequences of eigenfunctions. The resulting family of time-dependent Hamiltonians is such that, to the authors’ knowledge, have been unnoticed in the literature of stationary and nonstationary systems.

Singular dissipative third-order operator and its characteristic function

In this work, we describe well-defined dissipative boundary conditions related with a singular third-order differential equation in lim-3 case at singular point. Using the characteristic function of the corresponding dissipative operator we introduce a completeness theorem.

Third-order boundary value transmission problems

In this paper, we consider some third-order operators with transmission conditions. In particular, it is shown that such operators are formally symmetric in the corresponding Hilbert spaces and we introduce the resolvent operators associated with the differential operators. After showing that the eigenvalues of the problems are real and discrete we introduce some ordinary and Frechet derivatives of the eigenvalues with respect to some elements of data.

Third-order operators with three-point conditions associated with Boussinesq's equation

We consider a non-self-adjoint third-order operator on the interval with real 1-periodic coefficients and three-point Dirichlet conditions at the points 0, 1 and 2. The eigenvalues of this operator consist an auxiliary spectrum for the inverse spectral problem associated with the good Boussinesq equation. We determine eigenvalue asymptotics at high energy and the trace formula for the operator.

On the Completeness of a Part of Root Vectors for a Class of Third-Order Quasi-Elliptic Operator Pencils

On the Completeness of a Part of Root Vectors for a Class of Third-Order Quasi-Elliptic Operator Pencils

Interaction and energy transition between the breather and rogue wave for a generalized nonlinear Schrödinger system with two higher-order dispersion operators in optical fibers

A generalized nonlinear Schrodinger system is investigated, which can be used to describe the optical pulse propagation in inhomogeneous optical fibers with the fourth- and third-order dispersions operators. The Darboux transformation method is extended to construct a mixed breather and rogue wave solution for the system. The interaction behaviors between the breather and rogue wave are studied. As a novel result, the energy transition between the breather and rogue wave is observed. Furthermore, the impacts of the different operators on the mixed solution are analyzed.

On a class of third-order nonlocal Hamiltonian operators

Based on the theory of Poisson vertex algebras we calculate skew-symmetry conditions and Jacobi identities for a class of third-order nonlocal operators of differential-geometric type. Hamiltonian operators within this class are defined by a Monge metric and a skew-symmetric two-form satisfying a number of differential-geometric constraints. Complete classification results in the 2-component and 3-component cases are obtained.

Quantum dynamics of electron capture process during H–He2+ collision

Quantum dynamics of electron transfer (capture) phenomenon during the H–He2+ collision is investigated by solving two-dimensional time-dependent Schrodinger equation numerically using a third-order split operator technique. Results of this study, represented as the snapshots of the electron wavepacket time evolution, show significantly different dynamics for the electron of different initial orbitals (1s, 2s, 2px and 2py) of the incoming hydrogen atom. This electron transfer dynamics is also detailed by calculating expansion coefficients of the projection of the evolving wavepacket onto the stationary eigenfunctions of the H and He+ species to investigate evolution of the electron density around each nucleus during the collision. The instantaneous and overall electron densities captured by the He2+ nucleus from the H atom are calculated and analyzed. It is also shown evidently and concluded that due to its quantum nature, electron crawls from one nucleus to the other in an electron transfer process during an atom–ion collision.

On uniform Lebesgue constants of local exponential splines with equidistant knots

For a linear differential operator L r of arbitrary order r with constant coefficients and real pairwise different roots of the characteristic polynomial, we study Lebesgue constants (the norms of linear operators from C to C) of local exponential splines corresponding to this operator with a uniform arrangement of knots; such splines were constructed by the authors in earlier papers. In particular, for the third-order operator L 3 = D(D 2 − β 2) (β > 0), we find the exact values of Lebesgue constants for two types of local splines and compare these values with Lebesgue constants of exponential interpolation splines.