Exponential Spatial Decay Research Articles

Asymptotic Stability of Traveling Wave Solutions for Perturbations with Algebraic Decay

For a class of scalar partial differential equations that incorporate convection, diffusion, and possibly dispersion in one space and one time dimension, the stability of traveling wave solutions is investigated. If the initial perturbation of the traveling wave profile decays at an algebraic rate, then the solution is shown to converge to a shifted wave profile at a corresponding temporal algebraic rate, and optimal intermediate results that combine temporal and spatial decay are obtained. The proofs are based on a general interpolation principle which says that algebraic decay results of this form always follow if exponential temporal decay holds for perturbation with exponential spatial decay and the wave profile is stable for general perturbations.

Simultaneous single-frequency oscillations on different transitions in a laser-diode-pumped LiNdP_4O_12 laser

Simultaneous single-mode oscillations on different transitions were observed in a laser-diode-pumped microchip LiNdP(4)O(12) laser. The unusual modal-output characteristics of the laser were reproduced by multitransition-oscillation laser model equations that include cross saturation of population inversions with a strong spatial exponential decay and transition-dependent resonant reabsorption loss.

Localization of Mono‐ and Multichannel Waves in Disordered Periodic Systems

Analytical solution is obtained for the localization factor, namely, the exponential spatial decay rate, in the propagation of a disturbance through a chain of nearly periodic cell units. The localization is attributable to the departure from perfect spatial periodicity in the chain, known as disorder. In the case where multiple waves are permissible in the chain, each wave is characterized by a spatial Lyapunov exponent. These Lyapunov exponents constitute a spectrum of plus-minus pairs, and localization is governed by the largest negative Lyapunov exponent, which characterizes the least attenuated wave. This is then obtained by invoking Oseledec's multiplicative ergodic theorem, assuming that random disorders in different cell units are independent and identically distributed.

傾斜不均質な積層部からの弾性波反射

Discussion of elastic wave backscatter from an inhomogeneous layered part of a composite is made with the computational results by using a special layer elements. The stacking of linear inhomogeneous layer element, say LILE, has been certified as an excellent model for the computation of a gradient inhomogeneous layer. Intensity curves of the backscatter calculated as a function of incident wave number are presented in a wide frequency range, and characteristic relations between the inhomogeneous structure and the backscatter intensity are discussed. The results show that some peaks appear on the curves. Study of the characteristic features provides us viable clues for computation to get the information on layering structure and spatial exponential decay of inhomogeneous material constants of composites.

LOCALIZATION OF VIBRATION IN DISORDERED MULTI-SPAN BEAMS WITH DAMPING

The combined effects of disorder and structural damping on the dynamics of a multi-span beam with slight randomness in the spacing between supports are investigated. A wave transfer matrix approach is chosen to calculate the free and forced harmonic responses of this nearly periodic structure. It is shown that both harmonic waves and normal modes of vibration that extend throughout the ordered, undamped beam become spatially attenuated if either small damping or small disorder is present in the system. The physical mechanism which causes the attenuation, however, is one of energy dissipation in the case of damping but one of energy confinement in the case of disorder. The corresponding rates of spatial exponential decay are approximated by applying statistical perturbation methods. It is found that the effects of damping and disorder simply superpose for a multi-span beam with strong inter-span coupling, but interact less trivially in the weak coupling case. Furthermore, the effect of disorder is found to be small relative to that of damping in the case of strong inter-span coupling, but of comparable magnitude for weak coupling between spans. The adequacy of the statistical analysis to predict accurately localization in finite disordered beams with boundary conditions is also examined.

Theory and convergence properties of the screened Korringa-Kohn-Rostoker method.

Within the framework of the generalized multiple-scattering theory, a conceptually clear and transparent derivation of the real-space screened Korringa-Kohn-Rostoker method is presented. It is suggested that, by a suitable choice of the reference system, a fast exponential spatial decay of the structure constants can be obtained. This opens the way to treat large-scale systems in real space with a computational complexity that scales more favorably than the usual increase with the third power of the number of atoms.

Molecular dynamics simulations of nonequilibrium spatial correlations in a reaction diffusion system

The mesoscopic description of a system with chemical reactions predicts that if the detailed balance condition is not satisfied, then nonequilibrium spatial correlations between concentrations of reactants may appear. The present work is concerned with molecular dynamics simulations of such correlations in a model system of “reacting” hard spheres. The simulations have shown that if the detailed balance condition is satisfied, then the nonequilibrium spatial correlations are absent. In a steady state for which the detailed balance condition is not satisfied, the nonequilibrium correlations are present and their spatial exponential decay predicted by the theory based on a master equation, is confirmed in our simulations.

Vibration Confinement Phenomena in Disordered, Mono-Coupled, Multi-Span Beams

The effects of small, periodicity-destroying irregularities on the dynamics of multi-span beams are examined. It is shown that minute deviations of the span lengths from an ideal value alter qualitatively the dynamic response by localizing vibrations and waves to small geometric regions. These confinement phenomena are studied over a wide frequency range for beams resting on simple supports and with variable interspan coupling. Approximations of the localization factor (the average rate of spatial exponential decay of the vibration amplitude) are derived by statistical perturbation methods and validated by Monte Carlo simulations. If the interspan coupling is strong, localization effects are weak and of no practical significance for most engineering structures. On the other hand, localization is severe for small interspan coupling. Confinement is also generally stronger near the edges of frequency passbands and increases nearly linearly with frequency from passband to passband. This means that strong localization occurs at high frequencies, even for large static coupling between spans.

Some similarity temperature profiles for the microwave heating of a half-space

AbstractThere is presently considerable interest in the utilisation of microwave heating in areas such as cooking, sterilising, melting, smelting, sintering and drying. In general, such problems involve Maxwell's equations coupled with the heat equation, for which all thermal, electrical and magnetic properties of the material are nonlinear. The heat source arising from microwaves is proportional to the square of the modulus of the electric field intensity, and is known to increase with increasing temperature. In an attempt to find a simple model of microwave heating, we examine here simple transient temperature profiles corresponding to a heat source with spatial exponential decay but increasing with temperature, for which we assume either a power-law dependence or an exponential dependence. The spatial exponential decay is known to apply exactly when the electrical and magnetic properties of the material are assumed constant. A number of transient temperature profiles for this model are examined which arise from the invariance of the governing heat equation under simple one-parameter transformation groups. Some closed analytical expressions are obtained, but in general the resulting ordinary differential equations need to be solved numerically, and extensive numerical results are presented. For both models, these results indicate the appearance of moving fronts.

Scattering and decay theory for quaternionic quantum mechanics, and the structure of induced T nonconservation.

We develop a scattering theory and decay theory for nonrelativistic quaternionic quantum mechanics. We show that the far-zone part of scattering states lies in the complex openC(1,i) subspace of quaternionic Hilbert space picked out by the kinetic part of the Hamiltonian; intrinsically quaternionic terms are present in the wave function but have exponential spatial decay. Hence, scattering phase shifts are necessarily complex in quaternion quantum mechanics, and the test for quaternionic effects suggested by Peres gives a null result. Integrating out the quaternionic components, the complex scattering problem can be expressed in terms of an optical potential ${V}_{\mathrm{opt}}$ which is Hermitian but time-reversal nonconserving. The corresponding decay problem for openC(1,i) initial states can be expressed in terms of the same optical potential. Solving the decay problem to order ${V}_{\mathrm{opt}{}^{2}}$, the phenomenological form of the induced T nonconservation is seen to be ``milliweak,'' with T nonconservation arising both from the mass and the decay matrices, and hence is compatible with the phenomenology of T nonconservation in the standard model. When the complex openC(1,i) part of the quaternionic Hamiltonian has bound states, the optical potential develops isolated pole singularities. These lead to resonances at scattering energies equal to the bound-state binding energies, with widths proportional to the square of the quaternionic part of the potential. Inclusion of a positive-rest-mass term in the Hamiltonian shifts the location of these resonances towards lower kinetic energy, and if large enough eliminates them altogether.

On the Theory of Brillouin Scattering in Piezoelectric Semiconductors with Acoustic Phonon-Conduction Electron Interaction

Abstract The theory of nonresonant Brillouin scattering in anisotropic piezoelectric semiconductors with deformation potential coupling and piezoelectric coupling between excited systems of acoustic phonons and conduction electrons is reviewed. The scattering efficiency is calculated using the appropriate dyadic electromagnetic GreeN′s function. The depletion of the scattered intensity arising from a non phase-matched scattering kinematics and from a spatial exponential decay of the sound amplitude is taken into account. The contributions to the Brillouin scattering from the free-carrier-screened indirect photoelastic effect and from the free-carrier density modulation are expressed in terms of the self-consistent electric field. This field is obtained from a Boltzmann-equation calculation of the effective ac conductivity tensor. The acoustic dispersion of the Brillouin-scattering efficiency is considered, and some possibilities of determining electronic transport properties by means of Brillouin scatteri...

Asymptotic behaviour of eigenfunctions for multiparticle Schrödinger operators

O'Connor's approach to spatial exponential decay of eigenfunctions for multiparticle Schrodinger Hamiltonians is developed from the point of view of analytic perturbations with respect to transformation groups. This framework allows an improvement of his results in some directions; in particular if interactions are dilation analytic, exponential fall-off is shown to hold for any bound-state wave-function corresponding to an eigenvalue distinct from thresholds; it is shown that the exponential decay rate depends on the distance from the bound-state energy to the nearest threshold. Applications include non existence of positive energy bound-states.