Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide

We establish that by choosing a smooth local perturbation of the boundary of a planar quantum waveguide, we can create an eigenvalue near any given threshold of the continuous spectrum and the corresponding trapped wave exponentially decaying at infinity. Based on an analysis of an auxiliary object, a unitary augmented scattering matrix, we asymptotically interpret Wood’s anomalies, the phenomenon of fast variations in the diffraction pattern due to variations in the near-threshold wave frequency.     

Asymptotic behavior of the solution of singularly perturbed transmission problems in a periodic domain

This paper is devoted to the study of the asymptotic behavior of the solutions of singularly perturbed transmission problems in a periodically perforated domain. The domain is obtained by making in a periodic set of holes, each of them of size proportional to a positive parameter ε. We first consider an ideal transmission problem and investigate the behavior of the solution as ε tends to 0. In particular, we deduce a representation formula in terms of real analytic maps of ε and of some additional parameters. Then we apply such result to a nonideal nonlinear transmission problem.     

The discrete spectrum in the gaps of a second-order perturbed periodic operator

Leningrad State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 2, pp. 89–92, April–June, 1991.     

Eigenvalues in gaps of perturbed periodic Dirac operators: numerical evidence

This paper presents a method for the numerical investigation of the distribution of the eigenvalues introduced into a spectral gap of a periodic Dirac system by a perturbation of the type of the angular momentum term. A number of examples illustrate the effectiveness of the method and show the remarkable accuracy of the strong coupling asymptotic formula even for small values of the perturbation coupling constant. Furthermore, the results shed some light on the spectrum in the exceptional gap of radially periodic three-dimensional Dirac operators.     

Asymptotic behavior of solutions of a periodic diffusion system

This paper deals with a periodic reaction-diffusion system of plankton allelopathy under homogeneous Neumann boundary conditions. Based on the result of Ahmad and Lazer, we show some estimates and nonexistence results for the positive solutions of the system. Furthermore, we investigate the asymptotic behavior of the solutions of the system, that is one species dies out and the other exists as time t tends to infinity.     

Asymptotic representation of solutions of regularly perturbed systems of differential equations with nonclassical right-hand side

The problem of the B-asymptotic representation, with respect to a small parameter, of solutions of a regularly perturbed system of differential equations with impulse action on surfaces and of differential equations with discontinuous righthand side is considered. Theorems concerning the B-analytic dependence of solutions on the small parameter are proved. Algorithms for calculating the coefficients of the expansion are developed.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 43, No. 10, pp. 1298–1304, October, 1991.     

ASYMPTOTIC BEHAVIOR OF BOOTSTRAP SPECTRAL WINDOW ESTIMATION

ThisprojectissupportedbytheNationalNaturalScienceFoundationofChina.1.IntroductionLetX~{X(t);t=0,FI,12,'}beazerthmeantunitvariancestationaryGaussianprocess,theautocovariancefunction7(u)=EX(n u)X(n)satisfySupposethatthespectraldensityfunctionisf(A)andspectraldistributionfunctionisF(A)oftheprocessX,wherewerestrictAwithin11(if=[--x,7]).Fromtheassumptionsabove,wecaneasilyseethatthef(A)isjustaprobabilitydensityfunction,andF(A)isaprobabilitydistributionfunction.LetX(1),'tX(n)betheobserv…     

Asymptotic behavior for the singularly perturbed damped Boussinesq equation

This work is focused on the long‐time behavior of solutions to the singularly perturbed damped Boussinesq equation in a 3D case where ε > 0 is small enough. Without any growth restrictions on the nonlinearity f(u), we establish in an appropriate bounded phase space a finite dimensional global attractor as well as an exponential attractor of optimal regularity. The key step is the estimate of the difference between the solutions of the damped Boussinesq equation and the corresponding pseudo‐parabolic equation.Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic behavior of periodic dynamical on Banach spaces

Annali di Matematica Pura ed Applicata (1923 -) -     

Attractors and transients for a perturbed periodic KdV equation: A nonlinear spectral analysis

Summary In this paper we rigorously show the existence and smoothness inε of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parameterε. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions.     

Asymptotic behavior of solutions of a periodic diffusion system of plankton allelopathy

This paper is concerned with the existence and asymptotic behavior of periodic solutions for a periodic reaction diffusion system of a planktonic competition model under Dirichlet boundary conditions. The approach to the problem is by the method of upper and lower solutions and the bootstrap argument of Ahmad and Lazer. It is shown under certain conditions that this system has positive or semi-positive periodic solutions. A sufficient condition is obtained to ensure the stability and global attractivity of positive periodic solutions.     

Asymptotic behavior of solutions of nth-order ordinary differential equations with regularly varying nonlinearities

We obtain existence conditions and asymptotic, as t → ω (ω ≤ +∞), representations of a certain class of monotone solutions of an nth-order differential equation whose right-hand side is a sum of terms with regularly varying nonlinearities.     

Asymptotic behavior of the spectrum of a pseudodifferential operator with periodic bicharacteristics

Let _j be the eigenvalues of a positive elliptic pseudodifferential operator of order m > 0 on a closed compact d-dimensional C-manifold and let N()=#{j:_j^m}. It is shown that for each > 0 we have