Simple Eigenvalue Research Articles

A comparison of eigenvalue condition numbers for matrix polynomials

In this paper, we consider the different condition numbers for simple eigenvalues of matrix polynomials used in the literature and we compare them. One of these condition numbers is a generalization of the Wilkinson condition number for the standard eigenvalue problem. This number has the disadvantage of only being defined for finite eigenvalues. In order to give a unified approach to all the eigenvalues of a matrix polynomial, both finite and infinite, two (homogeneous) condition numbers have been defined in the literature. In their definition, very different approaches are used. One of the main goals of this note is to show that, when the matrix polynomial has a moderate degree, both homogeneous condition numbers are essentially the same and one of them provides a geometric interpretation of the other. We also show how the homogeneous condition numbers compare with the “Wilkinson-like” eigenvalue condition number and how they extend this condition number to zero and infinite eigenvalues.

Communication Using Eigenvalues of Higher Multiplicity of the Nonlinear Fourier Transform

A generalized Nonlinear Fourier Transform (GNFT), which includes eigenvalues of higher multiplicity, is considered for information transmission over fiber optic channels. Numerical algorithms are developed to compute the direct and inverse GNFTs. For closely-spaced eigenvalues, examples suggest that the GNFT is more robust than the NFT to the practical impairments of truncation, discretization, attenuation and noise. Communication using a soliton with one double eigenvalue is numerically demonstrated, and its information rates are compared to solitons with one and two simple eigenvalues.

Critical Domains for the First Nonzero Neumann Eigenvalue in Riemannian Manifolds

The present paper is devoted to geometric optimization problems related to the Neumann eigenvalue problem for the Laplace–Beltrami operator on bounded subdomains $$\Omega $$ of a Riemannian manifold $$({\mathscr {M}},g)$$. More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue $$\mu _2(\Omega )$$ with respect to volume-preserving domain perturbations, and we show that corresponding notions of criticality arise in the form of overdetermined boundary problems. Our results rely on an extension of Zanger’s shape derivative formula which covers the case when $$\mu _2(\Omega )$$ is not a simple eigenvalue. In the second part of the paper, we focus on product manifolds of the form $${\mathscr {M}}= {\mathbb {R}}^k \times {\mathscr {N}}$$, and we classify the subdomains where an associated overdetermined boundary value problem has a solution.

Dependence of eigenvalues on the diffusion operators with random jumps from the boundary

This paper deals with a non-self-adjoint eigenvalue problem{a(x)y″(x)+b(x)y′(x)=λy(x),y(0)=∫01y(x)dν0(x), y(1)=∫01y(x)dν1(x), which is associated with the generator of one dimensional diffusions with random jumps from the boundary. We focus on the dependence of spectral gap, eigenvalues and eigenfunctions on the coefficients a, b and the probability distributions ν0, ν1. To prove this, we show that all the eigenvalues are confined to a parabolic neighborhood of the real axis. Moreover, we also prove that zero is an algebraically simple eigenvalue of the problem.

Error Investigation of a Finite Element Approximation for a Nonlinear Sturm–Liouville Problem

A positive definite differential eigenvalue problem with coefficients depending nonlinearly on the spectral parameter is studied. The problem is formulated as a variational eigenvalue problem in a Hilbert space with bilinear forms depending nonlinearly on the spectral parameter. The variational problem has an increasing sequence of positive simple eigenvalues that correspond to a normalized system of eigenfunctions. The variational problem is approximated by a finite element mesh scheme on a uniform grid with Lagrangian finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions are proved depending on the mesh size and the eigenvalue size. The results obtained are generalizations of well-known results for differential eigenvalue problems with linear dependence on the spectral parameter.

Asymptotic stability for spectrally stable Lugiato-Lefever solitons in periodic waveguides

We consider the Lugiato-Lefever model of optical fibers in the periodic context. Spectrally stable periodic steady states were constructed recently in the studies of Delcey and Haragus [Philos. Trans. R. Soc., A 376, 20170188 (2018)]; [Rev. Roumaine Math. Pures Appl. (to be published)]; and Hakkaev et al. (e-print arXiv:1806.04821). The spectrum of the linearization around such solitons consists of simple eigenvalues 0, −2α < 0, while the rest of it is a subset of the vertical line {μ:Rμ=−α}. Assuming such a property abstractly, we show that the linearized operator generates a C0 semigroup and, more importantly, the semigroup obeys (optimal) exponential decay estimates. Our approach is based on the Gearhart-Prüss theorem, where the required resolvent estimates may be of independent interest. These results are applied to the proof of asymptotic stability with phase of the steady states.

Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: a unified framework

This paper develops a general framework for a posteriori error estimates in numerical approximations of the Laplace eigenvalue problem, applicable to all standard numerical methods. Guaranteed and computable upper and lower bounds on an arbitrary simple eigenvalue are given, as well as on the energy error in the approximation of the associated eigenvector. The bounds are valid under the sole condition that the approximate i-th eigenvalue lies between the exact $$(i-1)$$ -th and $$(i+1)$$ -th eigenvalue, where the relative gaps are sufficiently large. We give a practical way how to check this; the accuracy of the resulting estimates depends on these relative gaps. Our bounds feature no unknown (solution-, regularity-, or polynomial-degree-dependent) constant, are optimally convergent (efficient), and polynomial-degree robust. Under a further explicit, a posteriori, minimal resolution condition, the multiplicative constant in our estimates can be reduced by a fixed factor; moreover, when an elliptic regularity assumption on the corresponding source problem is satisfied with known constants, this multiplicative constant can be brought to the optimal value of 1 with mesh refinement. Applications of our framework to nonconforming, discontinuous Galerkin, and mixed finite element approximations of arbitrary polynomial degree are provided, along with numerical illustrations. Our key ingredients are equivalences between the i-th eigenvalue error, the associated eigenvector energy error, and the dual norm of the residual. We extend them in an appendix to the generic class of bounded-below self-adjoint operators with compact resolvent.

The existence of steady states for a bimolecular model with autocatalysis and saturation law

In this paper, a reaction–diffusion system known as a bimolecular model with autocatalysis and saturation law is considered. Firstly, we briefly obtain some characterizations for the positive solutions, including the a priori estimate of the positive solutions and the nonexistence of non-constant positive solution. Secondly, we emphatically discusses the bifurcation from the unique positive constant solution with both simple eigenvalues and double eigenvalues in one-dimensional case. Meanwhile, some other existence results are shown to supplement the analytical conclusions with the degree theory in N dimensional case.

The asymptotics of natural oscillations of a long two-dimensional Kirchhoff plate with variable cross-section

After scaling, a long Kirchhoff plate with rigidly clamped ends and free lateral sides is described by a mixed boundary-value problem for the biharmonic operator in a thin domain with weakly curved boundary. Based on a general procedure for constructing asymptotic formulae for solutions of elliptic problems in thin domains, asymptotic expansions are derived for the eigenvalues and eigenfunctions of this problem with respect to a small parameter equal to the relative width of the plate and are also justified. In the low-frequency range of the spectrum the limiting problem is a Dirichlet problem for a fourth-order ordinary differential equation with variable coefficients, and in the mid-frequency range it is (quite unexpectedly) a Dirichlet problem for a second-order equation. The phenomenon of a boundary layer in a neighbourhood of the ends of the plate is investigated. This makes it possible to construct infinite formal asymptotic series for simple eigenvalues and the corresponding eigenfunctions, and to develop a model with enhanced precision. Asymptotic constructions for plates with periodic rapidly oscillating boundaries or for other sets of boundary conditions corresponding to mechanically reasonable ways to fix the ends of the plate are discussed. Bibliography: 46 titles.

Finite Element Modeling of Eigenvibrations of a Loaded Bar

The nonlinear second-order differential eigenvalue problem describing eigenvibrations of a bar with elastically attached load is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. The sequence of eigenvalues corresponds to a system of normalized eigenfunctions. The initial nonlinear eigenvalue problem is approximated by the quadrature finite element method on a uniform grid. The existence and accuracy of approximate solutions are studied. Investigations of the present paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with elastically attached loads.

Spectrum theory of second-order difference equations with indefinite weight

In this paper, we study the spectrum structure of second-order difference operators with sign-changing weight. We apply the Sylvester inertia theorem to show that the spectrum consists of real and simple eigenvalues; the number of positive eigenvalues is equal to the number of positive elements in the weight function, and the number of negative eigenvalues is equal to the number of negative elements in the weight function. We also show that the eigenfunction corresponding to the j -th positive/negative eigenvalue changes its sign exactly j-1 times.

The Dirac Equation in the Kerr-de Sitter Metric

We consider a Fermion in the presence of a rotating BH immersed in a universe with a positive cosmological constant. After presenting a rigorous classification of the number and type of the horizons, we adopt the Carter tetrad to separate the aforementioned equation into radial and angular equations. We show how the Chandrasekhar ansatz leads to the construction of a symmetry operator that in the limit of a vanishing cosmological constant reproduces the square root of the squared total angular momentum operator for a Dirac particle in the Kerr metric. Furthermore, we prove that the the spectrum of the angular operator is discrete and consists of simple eigenvalues, and by means of the functional Bethe ansatz method we also derive a set of necessary and sufficient conditions for the angular operator to have polynomial solutions. Finally, we show that there exist no bound states for the Dirac equation in the non-extreme case.

On global bifurcation for a cross-diffusion predator–prey system with prey-taxis

This paper is concerned with a cross-diffusion predator–prey system with prey-taxis incorporating Holling type II functional response under homogeneous Neumann boundary condition. By employing global bifurcation theory, it is obtained that a branch of nonconstant solutions can bifurcate from the positive constant solution whenever the chemotactic is attractive or repulsive. Furthermore, by using perturbation of simple eigenvalues it is found that the bifurcating solutions are locally stable near the bifurcation point under suitable conditions. These results imply that cross-diffusion can create coexistence for the predators and preys under the above special case.

Asymptotics of resonant tunneling in a two-dimensional quantum waveguide with several equal resonators

ABSTRACTThe domain occupied by the waveguide is a strip with n+1 equal narrows of diameter ϵ. The wave function of a free electron satisfies the Dirichlet boundary value problem for the Helmholtz equation. Any part of the waveguide between two neighboring narrows plays the role of a resonator. Near a simple eigenvalue of the closed resonator, there are n resonant peaks of height close to 1. We let and obtain asymptotic formulas for the resonant values of the spectral parameter and for the widths of the resonant peaks at their half-height. The behavior of the transmission coefficient in a neighborhood of a resonance is described.

On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents

This study examines the instability and dynamical transitions of the two-layer western boundary currents represented by the Munk profile in the upper layer and a motionless bottom layer in a closed rectangular domain. First, a bound on the intensity of the Munk profile below which the western boundary currents are locally nonlinearly stable is provided. Second, by reducing the infinite dimensional system to a finite dimensional one via the center manifold reduction, non-dimensional transition numbers are derived, which determine the types of dynamical transitions both from a pair of simple complex eigenvalues as well as from a double pair of complex conjugate eigenvalues as the Reynolds number crosses a critical threshold. We show by careful numerical estimations of the transition numbers that the transitions in both cases are continuous at the critical Reynolds number. After the transition from a pair of simple complex eigenvalue, the western boundary layer currents turn into a periodic circulation, whereas a quasi-periodic or possibly a chaotic circulation emerges after the transition from a pair of double complex eigenvalues. Finally, a comparison between the transitions exhibited in one-layer and two-layer models is provided, which demonstrates the fundamental differences between the two models.

Zeno chattering of rigid bodies with multiple point contacts

Ideally rigid objects establish sustained contact with one another via complete chatter (a.k.a. Zeno behavior), i.e. an infinite sequence of collisions accumulating in finite time. Alternatively, such systems may also exhibit a finite sequence of collisions followed by separation (sometimes called incomplete chatter). Earlier works concerning the chattering of slender rods in two dimensions determined the exact range of model parameters, where complete chatter is possible. We revisit and slightly extend these results. Then the bulk of the paper examines the chattering of three-dimensional objects with multiple points hitting an immobile plane almost simultaneously. In contrast to rods, the motion of these systems is complex, nonlinear, and sensitive to initial conditions and model parameters due to the possibility of various impact sequences. These difficulties explain why we model this phenomenon as a nondeterministic discrete dynamical system. We simplify the analysis by assuming linearized kinematics, frictionless interaction, by neglecting the effect of external forces, and by investigating objects with rotational symmetry. Application and extension of the theory of common invariant cones of multiple linear operators enable us to find sufficient conditions of the existence of initial conditions, which give rise to complete chatter. Additional analytical and numerical investigations predict that our sufficient conditions are indeed exact, moreover solving a simple eigenvalue problem appears to be enough to judge the possibility of complete chatter. \keywords{contact dynamics \and chattering \and Zeno behavior \and common invariant cone

Stability of Bifurcating Stationary Solutions of the Artificial Compressible System

The artificial compressible system gives a compressible approximation of the incompressible Navier–Stokes system. The latter system is obtained from the former one in the zero limit of the artificial Mach number $$\epsilon $$ which is a singular limit. The sets of stationary solutions of both systems coincide with each other. It is known that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion, then it is also stable as a solution of the artificial compressible one for sufficiently small $$\epsilon $$ . In general, the range of $$\epsilon $$ shrinks when the spectrum of the linearized operator for the incompressible system approaches to the imaginary axis. This can happen when a stationary bifurcation occurs. It is proved that when a stationary bifurcation from a simple eigenvalue occurs, the range of $$\epsilon $$ can be taken uniformly near the bifurcation point to conclude the stability of the bifurcating solution as a solution of the artificial compressible system.

Krein signature for instability of [formula omitted]-symmetric states

Krein quantity is introduced for isolated neutrally stable eigenvalues associated with the stationary states in the PT-symmetric nonlinear Schrödinger equation. Krein quantity is real and nonzero for simple eigenvalues but it vanishes if two simple eigenvalues coalesce into a defective eigenvalue. A necessary condition for bifurcation of unstable eigenvalues from the defective eigenvalue is proved. This condition requires the two simple eigenvalues before the coalescence point to have opposite Krein signatures. The theory is illustrated with several numerical examples motivated by recent publications in physics literature.

Bifurcation results for a class of prescribed mean curvature equations in bounded domains

Bifurcation for prescribed mean curvature equations in one dimension has been intensively investigated in recent years, and a striking phenomenon discovered is that the length of the interval may affect the shapes of bifurcation curves. However, to our best knowledge, no such results are known in higher dimensions. In this paper, we study a class of prescribed mean curvature equations in bounded domains of RN with general nonlinearities f satisfying f(0)=0 and f′(0)>0. We establish some formulas for directions of bifurcation at simple eigenvalues, which lead to a sufficient and necessary condition to ensure that the directions depend on the size of the domain. In contrast, this phenomenon does not occur for the semilinear case. Some interesting examples, including logistic and perturbed exponential nonlinearities, are also investigated.

Spectra and Stability of Spatially Periodic Pulse Patterns II: The Critical Spectral Curve

In the stability analysis of pulse patterns in singularly perturbed reaction-diffusion systems, the scale separation is often exploited to reduce complexity. There are various methods to decompose the spectrum of the linearization about the pattern into slow and fast pieces and thus gain control over the spectrum in the asymptotic limit. These reduction methods have been developed in prototype models such as the FitzHugh--Nagumo, Gray--Scott, and Gierer--Meinhardt equations, which are of slowly linear nature; i.e., the slow dynamics away from the pulses are driven by linear equations. Recently, these methods were extended to homoclinic pulses in a general, slowly nonlinear class of reaction-diffusion systems. Yet, a straightforward extension to periodic pulse solutions is impossible due to two major obstructions. First, the reduction methods that lead to asymptotic spectral control for the homoclinic pulses fail for periodic patterns in slowly nonlinear systems. Second, there is a curve of spectrum attached to the origin that shrinks to the origin in the asymptotic limit, whereas for homoclinic pulses only a simple eigenvalue resides at the origin. Therefore, controlling the spectrum in the asymptotic limit is insufficient to decide upon stability. The first obstruction has been addressed in the companion paper [B. de Rijk, A. Doelman, and J. D. M. Rademacher, SIAM J. Math. Anal., 48 (2016), pp. 61--121]. In this paper we focus on the second obstruction: we obtain leading-order control over the critical spectral curve attached to the origin. The proof relies on Lin's method and utilizes exponential trichotomies. Our results yield explicit conditions for nonlinear stability and instability for stationary, spatially symmetric, periodic pulses in a large class of multicomponent, slowly nonlinear, reaction-diffusion systems. Moreover, we gain insight into the associated destabilization mechanisms.