On the instability of resonances in parallel-plane waveguides under local perturbations of the boundary

We study the large-time asymptotics for solutions u( x , t) of the wave equation with Dirichlet boundary data, generated by a time-harmonic force distribution of frequency ω, in a class of domains with non-compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω₀ ? ?² × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω₀. In particular, we prove that u is bounded as t → ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω₀ at the frequencies ω = πk (k = 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem ΔU + κ²U = ? f in Ω, U = 0 on ?Ω is reduced to a compact operator equation.     

On the essential spectrum of quantum waveguides

The main aim of the paper is the study of essential spectra of electromagnetic Schrödinger operators with variable potentials in cylindric domains , where is a bounded domain with a smooth boundary provided by admissible boundary conditions. Applying the limit operators method, we obtain explicit estimates of the essential spectrum for a wide class of quantum waveguides. We also consider a numerical example of calculations of the discrete spectrum of horizontally stratified quantum waveguides applying a method of the decomposition of solutions of spectral problems for one‐dimensional Schrödinger operators as a power series with respect to the spectral parameter. Copyright © 2012 John Wiley & Sons, Ltd.     

On the extrema of band functions in periodic waveguides

In the work we construct an example of a periodic differential operator whose spectrum has gaps with edges attained by the band functions at interior points of the Brillouin zone. This example is the Laplacian on a pair of infinite parallel strips with common boundary from which a periodic system of small holes is cut out. At that, on the outer boundaries the Dirichlet condition is imposed, while on the common boundary the Neumann condition is considered.     

On the Rayleigh-Plateau instability

In this paper,we study the surface instability of a cylindrical pore in the absence of stress. This instability is called the Rayleigh-Plateau instabilty. We consider the model developed by Spencer et ...     

An analytic estimation for the instability interval near second order resonances in the restricted three-body problem

Summary Periodic orbits of the first kind of the planar circular restricted three-body problem are linearly stable except in the neighbourhood of second order resonances, where an interval of unstable orbits appears when the mass parameter is not zero. Our main result is the estimation of the length of the instability interval.     

On a double cycle and resonances

We consider systems with 3/2 degrees of freedom close to nonlinear autonomous Hamiltonian ones in the case where the perturbed autonomous systems have a double limit cycle. Then the initial non-autonomous systems have a special resonance zone. The structure of this zone is investigated.     

On structures in instability zones

Two mathematical crystallization models describing structure formations in instability zones are proposed and justified. The first model, based on a phase field system, describes crystallization processes in binary alloys. The second model, based on a modified Biot model of a porous medium and the convective Cahn–Hilliard model, governs oriented crystallization. Physical interpretation and numerical analysis are discussed. Bibliography: 23 titles. Illustration: 5 figures.     

On resonances in systems of two weakly coupled oscillators

Assuming that two weakly coupled oscillators are essentially nonlinear we construct the most suitable form of a shortened 3-dimensional system which describes behavior of solutions inside non-degenerate resonance zones. We analyze a model system of that kind and establish the existence of limit cycles of different types and also the existence of nonregular attractors which are explained by the existence of saddle-focus loops.      

On the completeness of the system of eigenvectors of electromagnetic waveguides

Several statements of the spectral problem in the theory of regular waveguides are considered. The completeness of the system of eigenvectors of these problems, which is a consequence of the earlier established completeness property of a certain system of eigenvectors, is proved.     

On the instability theory of differential polynomials

Summary In this paper a class of n^th order non-linear differential equations is treated and solutions are sought which are asymptotically equivalent to logarithmic monomials. This paper is a shortened version of my doctoral dissertation submitted to Columbia University in May, 1964. I am especially grateful to ProfessorWalter Strodt who supervised the research, and whose advice was very helpful in the preparation of this paper. During a portion of the time this research was done,I was a National Science Foundation Cooperative Graduate Fellow.