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.     

The band spectrum of the periodic Airy–Schrödinger operator on the real line

We introduce the periodic Airy–Schrödinger operator and we describe its band spectrum. This is an example of solvable model with a periodic potential which is not differentiable at its extrema. We prove that there exists a sequence of explicit constants giving upper bounds of the semiclassical parameter for which explicit estimates are valid. We completely determine the behaviour of the edges of the first spectral band with respect to the semiclassical parameter. Then, we investigate the spectral bands and gaps situated in the range of the potential. We prove precise estimates on the widths of these spectral bands and these spectral gaps and we determine an upper bound on the integrated spectral density in this range. Finally, we get estimates of the edges of spectral bands and thus of the widths of spectral bands and spectral gaps which are stated for values of the semiclassical parameter in fixed intervals.     

Spectrum created by line defects in periodic structures

We study a Helmholtz‐type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a three‐dimensional periodic medium; the defect is infinitely extended in one direction, but compactly supported in the remaining two. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We will show that even small perturbations lead to additional spectrum in the spectral gaps of the unperturbed operator and investigate some properties of the spectrum that is created.     

Asymptotics of bound states and bands for waveguides coupled through small windows

A system of two waveguides coupled laterally through small windows is considered. The asymptotics (in the width of windows) of ground state close to the threshold is obtained for the case of finite number of apertures. The cases of periodic system of coupling windows is studied. The asymptotics of the band edges is obtained. The technique is matching of asymptotic expansions of the solutions.     

Coupled dielectric waveguides with photonic crystal properties

A system of planar waveguides coupled through a periodic set of small windows is considered. It is shown that the weak periodic coupling of the waveguides leads to an additional eigenvalue band separated by a gap from the threshold of a continuous spectrum branch. Thus, the system has photonic crystal properties, which can be used to construct optical fiber devices. This system can also play the role of a SCISSOR device for reducing the group velocity of light, in optical delay lines, etc.     

Quasiconformal mappings and periodic spectral problems in dimension two

We study spectral properties of second-order elliptic operators with periodic coefficients in dimension two. These operators act in periodic simply-connected waveguides, with either Dirichlet, or Neumann, or the third boundary condition. The main result is the absolute continuity of the spectra of such operators. The cornerstone of the proof is an isothermal change of variables, reducing the metric to a flat one and the waveguide to a straight strip. The main technical tool is the quasiconformal variant of the Riemann mapping theorem. This work is supported by The Royal Society.     

The localization for eigenfunctions of the dirichlet problem in thin polyhedra near the vertices

Under some geometric assumptions, we show that eigenfunctions of the Dirichlet problem for the Laplace operator in an n-dimensional thin polyhedron localize near one of its vertices. We construct and justify asymptotics for the eigenvalues and eigenfunctions. For waveguides, which are thin layers between periodic polyhedral surfaces, we establish the presence of gaps and find asymptotics for their geometric characteristics.     

The rigorous derivation of the T2 focusing cubic NLS from 3D

We derive rigorously the 2D periodic focusing cubic NLS as the mean-field limit of the 3D focusing quantum many-body dynamics describing a dilute Bose gas with periodic boundary condition in the x-direction and a well of infinite-depth in the z-direction. Physical experiments for these systems are scarce. We find that, to fulfill the empirical requirement for observing NLS dynamics in experiments, namely, that the kinetic energy dominates the potential energy, it is necessary to impose an extra restriction on the system parameters. This restriction gives rise to an unusual coupling constant.     

Asymptotic behavior of spectral gaps in a regularly perturbed periodic waveguide

Since the spectrum of a periodic waveguide is the union of a countable family of closed bounded segments (spectral bands), it can contain opened spectral gaps, i.e., intervals in the real positive semi-axis that are free of the spectrum but have both endpoints in it. A cylindrical waveguide has an intact spectrum that is a closed ray. We consider a small periodic perturbation of the waveguide wall, and, by means of an asymptotic analysis of the eigenvalues in the model problem on the periodicity cell, we show how a spectral gap opens when the cylindrical waveguide converts into a periodic one. Indeed, a cylindrical waveguide can be interpreted as a periodic one with an arbitrary period, but all its spectral bands touch each other. A periodic perturbation of the waveguide wall provides the splitting of the band edges. This effect is known in the physical literature for waveguides of different shapes, and, in this paper, we provide a rigorous mathematical proof of the effect. Several variants of the edge splitting (alone and coupled, simple and multiple knots) are examined. Explicit formulas are obtained for a plane waveguide.     

Opening gaps in the spectrum of the water-wave problem in a periodic channel

It is shown that the essential spectrum of a problem in the theory of linear water waves in a periodic channel can contain any prescribed number of gaps. One of such waveguides consists of a periodic family of identical ponds of unit size connected by narrow shallow channels. The effect of gap opening is achieved by decreasing a geometric parameter describing the size of these channels.     

Replicate periodic windows in the parameter space of driven oscillators

In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have the same periodicity and pattern of stable and unstable periodic orbits already existent for the unperturbed oscillator. Moreover, these unstable periodic orbits are embedded in chaotic attractors in phase space regions where the new stable orbits are identified. Thus, the observed periodic window replication is an effective oscillator control process, once chaotic orbits are replaced by regular ones.     

High-order approximations for an incompressible viscous flow on a rough boundary

In this paper, we propose approximations of fluid flow that could be used for obtaining wall laws of higher order. We consider the two-dimensional laminar fluid flow, modeled by the incompressible Stokes system in a straight channel with a rough side. The roughness is periodic and the ratio of the amplitude of the rough part and the size of the flow domain is denoted by ?, being a small number. We impose periodic boundary conditions on the flow. We generalize the boundary layers needed for the construction of flow approximations of higher order with respect to ?. The existence of the layers and their features are discussed. Finally we give the error estimates for the approximations and establish an explicit wall law.     

Uniform resolvent convergence for strip with fast oscillating boundary

In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary we impose Dirichlet, Neumann or Robin boundary condition. In all cases we describe the homogenized operator, establish the uniform resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. These results are obtained as the order of the amplitude of the oscillations is less, equal or greater than that of the period. It is shown that under the homogenization the type of the boundary condition can change.     

On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition

We consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resolvent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results are the uniform resolvent convergence for the operator on the cell of periodicity obtained by the Floquet–Bloch decomposition, the two terms asymptotics for the band functions, and the complete asymptotic expansion for the bottom of the spectrum with an exponentially small error term.     

Asymptotics and Estimates of the Convergence Rate in a Three-Dimensional Boundary-Value Problem with Rapidly Alternating Boundary Conditions

We consider a singularly perturbed boundary-value eigenvalue problem for the Laplace operator in a cylinder with rapidly alternating type of the boundary condition on the lateral surface. The change of the boundary conditions is realized by splitting the lateral surface into many narrow strips on which the Dirichlet and Neumann conditions alternate. We study the case in which the averaged problem contains the Dirichlet boundary condition on the lateral surface. In the case of strips with slowly varying width we construct the first terms of the asymptotic expansions of eigenfunctions; moreover, in the case of strips with rapidly varying width we obtain estimates for the convergence rate.     

Periodic segment implies infinitely many periodic solutions

In this note we show that the existence of a periodic segment for a non-autonomous ODE with periodic coefficients implies the existence of infinitely many periodic solutions inside this segment provided that a sequence of Lefschetz numbers of iterations of an associated map is not constant. In the case when this sequence is bounded we have to impose a geometric condition on the segment to get solutions by use of symbolic dynamics.

     

A gap in the spectrum of the Neumann–Laplacian on a periodic waveguide

We study a spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder which contains a periodic arrangement of inclusions. On the boundary of the waveguide, we consider both Neumann and Dirichlet conditions. We prove that provided the diameter of the inclusion is small enough the spectrum of Laplace operator contains band gaps, i.e. there are frequencies that do not propagate through the waveguide. The existence of the band gaps is verified using the asymptotic analysis of elliptic operators.     

A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition

We study a free boundary problem for the Laplace operator, where we impose a Bernoulli-type boundary condition. We show that there exists a solution to this problem. We use A. Beurling’s technique, by defining two classes of sub- and super-solutions and a Perron argument. We try to generalize here a previous work of A. Henrot and H. Shahgholian. We extend these results in different directions.     

The Poisson limit theorem for high extrema of a time series with seasonal component and monotone trend

Poisson’s limit theorem is considered in the paper for high extrema of a stationary time series with a monotone trend and an almost periodic component. It is assumed that the distribution function of the time series is maximum stable and the time series satisfies the weak dependence condition. The limit behavior of the random process of high extrema for this model is considered for the first time.     

On Initial-Boundary Value Problem of the Stochastic Heat Equation in Lipschitz Cylinders

We consider the initial boundary value problem of the non-homogeneous stochastic heat equation. The derivative of the solution with respect to time receives heavy random noises. The space boundary is Lipschitz, and we impose nonzero condition on the parabolic boundary. We prove a regularity result by finding appropriate spaces for solutions and pre-assigned data in the problem. We use a collection of tools from potential theory, harmonic analysis, and probability. Some lemmas are as important as the main theorem.