Nonzero Real Parts Research Articles

Eigenvalue Analysis and Longtime Stability of Resonant Structures for the Meshless Radial Point Interpolation Method in Time Domain

A meshless collocation method based on radial basis function (RBF) interpolation is presented for the numerical solution of Maxwell's equations. RBFs have attractive properties such as theoretical exponential convergence for increasingly dense node distributions. Although the primary interest resides in the time domain, an eigenvalue solver is used in this paper to investigate convergence properties of the RBF interpolation method. The eigenvalue distribution is calculated and its implications for longtime stability in time-domain simulations are established. It is found that eigenvalues with small, but nonzero, real parts are related to the instabilities observed in time-domain simulations after a large number of time steps. Investigations show that by using global basis functions, this problem can be avoided. More generally, the connection between the high matrix condition number, accuracy, and the magnitude of nonzero real parts is established.

Multiscale analysis for diffusion-driven neutrally stable states

The Turing instabilities for reaction–diffusion systems are studied from the Fourier normal modes which appear by searching the solution obtained from linearization of the reaction–diffusion system at the spatially homogeneous steady state. The linear stability analysis is only appropriate when the temporal eigenvalues associated to every given spatial eigenvalue have non-zero real part. If the real part of the temporal eigenvalue in a normal mode is equal to zero there is no enough information coming from the linearized system. Given an arbitrary spatial eigenvalue, by equating to zero the real part of the corresponding temporal eigenvalue will lead to a neutral stability manifold in the parameter space. If for a given spatial eigenvalue the other parameters in the reaction–diffusion process drive the system to the neutral manifold, then neither stability nor instability can be warranted by the usual linear analysis. In order to give a sketch of the nonlinear analysis we use a multiple scales method. As an application, we analyze the behavior of solutions to the Schnakenberg trimolecular reaction kinetics in the presence of diffusion.

A numerical method for quadratic eigenvalue problems of gyroscopic systems

We consider the quadratic eigenvalues problem (QEP) of gyroscopic systems ( λ 2 M + λ G + K ) x = 0 , where M = M ⊤ , G = - G ⊤ and K = K ⊤ ∈ R n × n with M being positive definite. Guo [Numerical solution of a quadratic eigenvalue problem, Linear Algebra and its Applications 385 (2004) 391–406] showed that all eigenvalues of the QEP can be found by solving the maximal solution of a nonlinear matrix equation Z + A ⊤ Z - 1 A = Q with quadratic convergence when the QEP has no eigenvalues on the imaginary axis. The convergence becomes linear or more slower (Guo, 2004) when the QEP allows purely imaginary eigenvalues having even partial multiplicities. In this paper, we consider the general case when the QEP has eigenvalues on the imaginary axis. We propose an eigenvalue shifting technique to transform the original gyroscopic system to a new gyroscopic system, which shifts purely imaginary eigenvalues to eigenvalues with nonzero real parts, while keeps other eigenpairs unchanged. This transformation ensures that the new method for computing the maximal solution of the nonlinear matrix equation converges quadratically. Numerical examples illustrate the efficiency of our method.

Model of light collimation by photonic crystal surface modes

We propose a quantitative model explaining the mechanism of light collimation by leaky surface modes that propagate on a corrugated surface around the output of a photonic crystal waveguide. The dispersion relation of these modes is determined for a number of surface terminations. Analytical results obtained on the basis of the model are compared to those of rigorous numerical simulations. Maximum collimation is shown to occur at frequency values corresponding to excitation of surface modes whose wave number retains a nonzero real part.

Structurally chiral photonic crystals with magneto-optic activity: indirect photonic bandgaps, negative refraction, and superprism effects

We have theoretically investigated the general properties of photonic crystals without time-reversal and space-inversion symmetries by examining the case of one-dimensional periodic, lossless dielectric helical media (HM) with magneto-optic (MO) activity. We show that photonic bandgap formation in the absence of these two symmetry elements leads to a remarkable set of properties: indirect photonic bandgaps (edges not aligned in k space), backward wave propagating eigenmodes (which allow for negative refraction), unusual nonpropagating modes in the gap (their complex-valued wave vectors have frequency-dependent, nonzero real parts, which do not change sign for opposite decay directions), anomalous propagation, and group-velocity-based unidirectional superprism effects. Particular properties of MO HM are discussed in detail.

Eigenvalues of the Zakharov-Shabat scattering problem for real symmetric pulses.

The classical problem of determining the solitons generated from symmetric real initial conditions in the nonlinear Schrödinger equation is revisited. The corresponding Zakharov-Shabat scattering problem is solved for real and symmetric double-humped rectangular initial pulse forms. It is found that such real symmetric pulses may generate eigenvalues with nonzero real parts corresponding to separating soliton pulse pairs. Moreover, it is found that the classical formula relating the number of eigenvalues to the area of the pulse is not always correct.

Trichotomies for abstract semilinear differential equations

In this paper we present some results concerning the existence and uniqueness of mild solutions to certain abstract semilinear differential equations and the asymptotic behavior of these solutions. The basic techniques used are the iterative method and the fixed point theory for differential equations in Banach space. However, the most pleasant here is that it can be applied to nonlinear equations without assuming the eigenvalues of the differential operator in the linear parts of the differential equation has non-zero real part.

Anomalous absorption of electromagnetic waves in a layered structure with complex permittivities

It is shown theoretically that the absorptivity of bulk TE and TM electromagnetic waves by a thin layer with large complex dielectric permittivity has a strong (as much as 100%) maximum at an optimal layer thickness that is usually much smaller than the wavelength and the wave penetration depth (skin-depth) in the layer. The analysis is carried out in the frictional contact approximation, in which the layer can be replaced by a contact of the surrounding media with special boundary conditions that are analogous to the conditions at a contact with linear mechanical friction. The effects of a non-zero real part of the layer permittivity, and non-zero dissipation in the surrounding media on the anomalous absorption are investigated theoretically in detail. Simple analytical equations for the absorptivity maximum, optimal layer thickness and optimal incidence angles are derived and analysed. Differences in the anomalous absorption of TE and TM waves in layered structures with complex permittivities are described and explained from the physical point of view.

Diffraction of a sound wave on an open end of a half-infinite waveguide with impedance walls and impedance flanges

Diffraction of a plane sound wave on an open end of a waveguide with impedance walls, connected with impedance flanges, is considered. A plane wave is incident upon the waveguide from open space. As a result of diffraction, part of the sound energy is scattered in the half-space and the other part of the energy penetrates into the waveguide. Results of the solution of this problem can be used for calculation of the amplitude of a sound wave, scattered by a hole, in particular, in the backward direction. Besides, one can calculate the sound energy penetrating into the waveguide. Absorption of the energy in the waveguide can be taken into account by introducing impedances with nonzero real parts. The distortion of the directivity pattern of a system of receivers located in the waveguide can also be obtained. The solution of the problem has been carried out by the integral equation method using Green’s function of the half-space with an impedance boundary. The equation was solved by expanding an unknown function in the eigenfunctions of the waveguide. It reduces to an infinite set of linear algebraic equations. Results of computation of bistatic diagrams of scattering on the hole and diagrams of backward scattering are presented.

The behaviour of the propagating and evanescent modes of a 2D quantum wire in a perpendicular magnetic field

The modes of a 2D quantum wire in a perpendicular magnetic field are studied in detail for the real electron energies which are appropriate in nanostructure scattering problems. Propagating and evanescent modes are considered for both asymmetric and symmetric confining potentials. The symmetry of the mode functions and the real and imaginary parts ofkare investigated in both cases. For asymmetric potentials the evanescentkvalues usually have non-zero real parts. For symmetric potentials purely imaginarykvalues also appear when ϵ is just below a propagation threshold. Calculated wavefunctions and dispersion curves for both Re(k) and Im(k) are presented for simple examples of both cases and the transformation from one to the other is described. The total current carried by an arbitrary superposition of modes (both propagating and evanescent) is calculated. The part of the current carried by interfering evanescent modes involves a complex analogue of the group velocity.

Complex Periodic Orbits and Tunneling in Chaotic Potentials.

We derive a trace formula for the splitting-weighted density of states suitable for chaotic potentials with isolated symmetric wells. This formula is based on complex orbits which tunnel through classically forbidden barriers. The theory is applicable whenever the tunneling is dominated by isolated orbits, a situation which applies to chaotic systems but also to certain near-integrable ones. It is used to analyze a specific two-dimensional potential with chaotic dynamics. Mean behavior of the splittings is predicted by an orbit with imaginary action. Oscillations around this mean are obtained from a collection of related orbits whose actions have nonzero real part. {copyright} {ital 1996 The American Physical Society.}

FAMILIES OF TRANSVERSE POINCARÉ HOMOCLINIC ORBITS IN 2N-DIMENSIONAL HAMILTONIAN SYSTEMS CLOSE TO THE SYSTEM WITH A LOOP TO A SADDLE-CENTER

We prove the theorem: if an n-degrees-of-freedom Hamiltonian system has an equilibrium of the saddle-center type (there is a pair of simple eigenvalues ±iω; the rest of the spectrum consists of eigenvalues with nonzero real parts) with a homoclinic orbit to it then this system, and all those close to it, have transversal Poincaré homoclinic orbits to Lyapunov periodic orbits if some genericity conditions are satisfied. These conditions are pointed out explicitly. Thus a new criterion of nonintegrability has been obtained.

Relaxing the detectability condition for the algebraic Riccati equation

For a linear system, the detectability condition can be weakened to the Hamiltonian matrix composed of (A, B, Q, R) having eigenvalues with nonzero real parts, for the existence of the unique positive semidefinite stabilising solution of the algebraic Riccati equation. This condition yields the optimal feedback gain and can always be easily obtained by a given iterative technique.

Some properties of elastodynamic eigensolutions in stratified media

In this note, we consider some symmetries of the coefficient matrix in the differential system for the displacement-stress vector in stratified media. These symmetries are used to obtain extremely simple relations between the elements of an associated eigenvector matrix and the elements of the inverse of this matrix. These results obtain in lossless but otherwise arbitrary elastic solids and, in particular, they hold for anisotropic media, obviating numerical calculation of the inverse eigenvector matrix in this case. The relations are closely tied to the notion of vertical energy flux normalization for eigensolutions associated with pure imaginary eigenvalues, but a different normalization naturally arises when the eigenvalues have a non-zero real part. The new normalization can be easily related to vertical energy flux normalization for travelling waves.

Some properties of elastodynamic eigensolutions in stratified media

Summary. In this note, we consider some symmetries of the coefficient matrix in the differential system for the displacement-stress vector in stratified media. These symmetries are used to obtain extremely simple relations between the elements of an associated eigenvector matrix and the elements of the inverse of this matrix. These results obtain in lossless but otherwise arbitrary elastic solids and, in particular, they hold for anisotropic media, obviating numerical calculation of the inverse eigenvector matrix in this case. The relations are closely tied to the notion of vertical energy flux normalization for eigensolutions associated with pure imaginary eigenvalues, but a different normalization naturally arises when the eigenvalues have a non-zero real part. The new normalization can be easily related to vertical energy flux normalization for travelling waves. The equations of motion for linear elastodynamics are separable in a Cartesian coordinate system when the elastic properties and the density of the medium depend on a single coordinate, say x3 = z. In this case, a general time-dependent displacement may be resolved into harmonic wavetrains of the form u = u(z) exp[i(kl x, t k2 x2 - wt)].

Reflection behavior of linear slip interfaces

A linear slip interface is one across which tractions are continuous but normal or shear displacements may be discontinuous [M. Schoenberg, J. Acoust. Soc. Am. 66, S2 (1979)]. At low frequencies, the slip interface is equivalent to a thin low impedance layer. If the compliance has nonzero real part, it will produce a frequency‐dependent reflection coefficient. In this paper, we describe the reflection behavior of a system consisting of an acoustic half‐space, an elastic layer, a slip interface, and an elastic half‐space. The frequency response of this system is affected by both the thickness resonance of the elastic layer and the compliance of the slip interface. Comparisons are made with a similar system lacking the slip interface and with a system in which the slip interface is replaced by a thin low impedance layer.

A sufficient condition that a matrix have eigenvalues with non-zero real parts

If A is a complex matrix let A be the real matrix obtained by replacing the diagonal elements of A by the moduli of their real parts and by replacing the off-diagonal elements by the negative of their moduli. Then we show that if A is an M-matrix the eigenvalues of A have nonzero real parts and, moreover, the moduli of the real parts are bounded below by the minimum of the real parts of the eigenvalues of A.

A diagonal dominance criterion for exponential dichotomy

Roughly speaking, a system of linear differential equations has an exponential dichotomy if it has a subspace of solutions shrinking exponentially and a complementary subspace of solutions growing exponentially. In the case of constant coefficients, this happens if and only if the eigenvalues of the coefficient matrix have nonzero real parts. In the general case, Lazer has shown that if the coefficient matrix function is bounded and satisfies a diagonal dominance condition (which, in the constant case, is a sufficient but not necessary condition that the eigenvalues have nonzero real parts) then the system has an exponential dichotomy. In this paper we prove the same result with a weaker diagonal dominance condition, thus generalizing a theorem of Nakajima.

Complex local equivalents to imaginary non-local potentials: Neutron scattering from 208Pb

The complex equivalent local potential to an imaginary non-local potential obtained in a weak coupling particle vibration model for low energy s-wave neutron elastic scattering from 208Pb is presented. The local potential contains a non-zero real part, is surface peaked, and exhibits resonances. The equivalent local potential and damping of the local wave function relative to the non-local wave function depend on the strength of the Woods-Saxon real central potential also present in the Schroedinger equation.

On the existence of periodic solutions on 2-manifolds

In this paper we shall study flows generated by a Cl vector field on a compact two-dimensional manifold. For the sake of simplicity we shall assume that the manifold is orientable in which case the manifold is P, a sphere with h handles. We shall assume that the vector field has a finite, positive number of nondegenerate singularities. Here, “nondegenerate” is taken to mean that the eigenvalues associated with the linearized vector field at a singularity have nonzero real parts. The problem that then interests us is to determine when there exists a nontrivial periodic motion on .?Yh. (A singularity is defined to be a trivial periodic motion.)