Study at high frequencies of a stratified waveguide

A waveguide in integrated optics is defined by its refractiveindex. The guide is assumed to be invariant in the propagationdirection while in the transverse direction it is supposed tobe a compact perturbation of an unbounded stratified medium.We are interested in the high frequency modes guided by thisdevice. We consider the problem under the assumptions of weak guidance,so that it reduces to a two-dimensional eigenvalue problem fora scalar field. While a general study has been done in a previouspaper (Bonnet-BenDhia et al., IMAJAM 60, 1998), our goal hereis to present an asymptotic study at high frequencies, whichillustrates the dispersive character of the stratified guide.We will give the limit as the frequency tends to of the guidedmodes and characterize this limit as the solution of an eigenproblem.The technical difficulty lies in the stratified character ofthe unbounded reference medium.     

A global existence result for a spatially extended 3D Navier-Stokes problem with non-small initial data

We consider the Navier-Stokes equations in a two- or three-dimensional unbounded cylindrical domain. The existence and uniqueness of solutions is discussed in the space of uniformly local square integrable functions. We show for small initial data and small forcing term that the solutions exist globally in time. This result is extended to a non-small data result in the sense that the high frequency modes of the initial conditions and of the forcing terms are allowed to be large. Moreover, we show the existence of a local attractor for this 3D Navier-Stokes problem in an unbounded domain. In contrast to previous results the spaces used are no Hilbert spaces, and secondly we have a linear operator possessing continuous spectrum without spectral gap. Received March 1999     

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.     

Oscillations of stratified fluids

We study the problem on small motions and normal oscillations of a system of two heavy immiscible stratified fluids partially filling a fixed vessel. The lower fluid is assumed to be viscous, while the upper one is assumed to be ideal. We find sufficient existence conditions for a strong (with respect to the time variable) solution of the initial-boundary value problem describing the evolution of the specified hydraulic system. For the corresponding spectral system, we obtain results about the localization of the spectrum, asymptotic behavior of branches of eigenvalues, and existence of the substantial spectrum of the problem.     

On the existence of infinitely many nonperturbative solutions in a transmission eigenvalue problem for nonlinear Helmholtz equation with polynomial nonlinearity

The paper focuses on a transmission eigenvalue problem for nonlinear Helmholtz equation with polynomial nonlinearity which describes the propagation of transverse electric waves along a dielectric layer filled with nonlinear medium. It is proved that even if the nonlinearity coefficients are small, the nonlinear problem has infinitely many nonperturbative solutions, whereas the corresponding linear problem always has a finite number of solutions. This results in the theoretical existence of a novel type of nonlinear guided waves that exist only in nonlinear guided systems. Asymptotic distribution of the eigenvalues is found and a comparison theorem is proved; periodicity of the eigenfunctions is proved, the exact formula for the period is found, and the zeros of the eigenfunctions are determined. The results found essentially extend the theory evolved earlier (particular cases for Kerr, cubic-quintic, septic nonlinearities, etc. are easily extracted from the general results found here). Numerical results are also presented.     

Some Sharp Norm Estimates in the Subspace Perturbation Problem

We discuss the spectral subspace perturbation problem for a self-adjoint operator. Assuming that the convex hull of a part of its spectrum does not intersect the remainder of the spectrum, we establish an a priori sharp bound on variation of the corresponding spectral subspace under off-diagonal perturbations. This bound represents a new, a priori, tan Θ Theorem. We also extend the Davis–Kahan tan 2Θ Theorem to the case of some unbounded perturbations.     

Guided waves in a stratified elastic and locally perturbed domain: theoretical and numerical aspects

This article presents new results concerning guided waves in a three‐dimensional unbounded stratified and locally perturbed elastic medium. On the one hand, we numerically show non‐monotonic dispersion curves, a phenomenon not yet encountered in fields other than elasticity. On the other hand, we prove that the infimum of the essential spectrum of the studied operator depends on the possible non‐monotonicity of such curves. This link is a new result with respect to equivalent situations coming from acoustics or electromagnetism. The numerical study underlines, besides the non‐monotonic dispersion curves, the appearance of generalized Stoneley waves. Copyright © 2000 John Wiley & Sons, Ltd.     

Analysis of the spectrum of a Cartesian Perfectly Matched Layer (PML) approximation to acoustic scattering problems

In this paper, we study the spectrum of the operator which results when the Perfectly Matched Layer (PML) is applied in Cartesian geometry to the Laplacian on an unbounded domain. This is often thought of as a complex change of variables or “complex stretching.” The reason that such an operator is of interest is that it can be used to provide a very effective domain truncation approach for approximating acoustic scattering problems posed on unbounded domains. Stretching associated with polar or spherical geometry lead to constant coefficient operators outside of a bounded transition layer and so even though they are on unbounded domains, they (and their numerical approximations) can be analyzed by more standard compact perturbation arguments. In contrast, operators associated with Cartesian stretching are non-constant in unbounded regions and hence cannot be analyzed via a compact perturbation approach. Alternatively, to show that the scattering problem PML operator associated with Cartesian geometry is stable for real nonzero wave numbers, we show that the essential spectrum of the higher order part only intersects the real axis at the origin. This enables us to conclude stability of the PML scattering problem from a uniqueness result given in a subsequent publication.     

Spectre essentiel et ondes élastiques guidées dans l'espace stratifié et perturbé

In this Note we point out the consequences of the existence of guided waves in a tridimensional, stratified, locally disturbed, elastic medium that is invariant by translation in the direction x₃, and unbounded. The main result concerns the lower value of the essential spectrum, this value is obtained from the dispersion curve of the first eigenvalue σ_p,1 (βf) of the reduced operator A_p(β); however, theses values are not necessarily equal as it is the case in electromagnetism and acoustic. It is linked with the aspect of the dispersion curves of the plane associated operator, curves which are not necessarily monotone and we have inf σ_ess,(A(β)) = min_ε≥(σ_p,1 (ξ)).     

Propagation dynamics of a nonlocal spatial Lyme disease model in a time–space periodic habitat

This paper is concerned with the investigation of Lyme disease spread via a time–space periodic nonlocal spatial model in an unbounded domain. We first study the spatial periodic initial problem of the model system and discuss the existence of principal eigenvalue of a linear system with the spatial nonlocality induced by time delay under a smooth assumption. Then we establish the existence of the spreading speeds, and show its coincidence with the minimal wave speed. We further perform a perturbation argument to remove this aforementioned assumption and provide an estimation of the spreading speeds in terms of the spectral radius. Simulations are presented to illustrate our analytic results.     

New results for guided waves in heterogeneous elastic media

We prove the existence of guided waves propagating with a velocity strictly larger than the S (shear) wave velocity at infinity in the case of unbounded elastic media invariant under translation in one space direction and asymptotically homogeneous at infinity. These waves correspond to the existence of eigenvalues embedded in the essential spectrum of the self-adjoint elastic propagator.     

Dichotomy,bisectorial operators and unbounded spectral projections

For an operator having a uniformly bounded resolvent on a strip around the imaginary axis, the existence of—possibly unbounded—spectral projections corresponding to the left and right half-plane is proved. The operator is dichotomous if these projections are bounded, and an abstract perturbation theorem for dichotomy is derived. All results apply, with certain simplifications, to bisectorial operators. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Evolution of Linearized Perturbations of Parallel Flows

The equations of an incompressible fluid are linearized for small perturbations of a basic parallel flow. The initial-value problem is then posed by use of Fourier transforms in space. Previous results are systematized in a general framework and used to solve a series of problems for prototypical examples of basic shear flow and of initial disturbance. The prototypes of shear flow are (a) plane Couette flow bounded by rigid parallel walls, (b) plane Couette flow bounded by rigid walls at constant pressure, (c) unbounded two-layer flow with linear velocity profile in each layer, (d) a piecewise linear profile of a boundary layer on a rigid wall. The prototypes of initial perturbation are the fundamental ones: (i) a point source of the field of the transverse velocity (represented by delta functions), (ii) an unbounded sinusoidal field of the transverse velocity, (iii) a point source of the lateral component of vorticity, (iv) a sinusoidal field of the lateral vorticity. Detailed solutions for an inviscid fluid are presented, but the problem for a viscous fluid is only broached.     

On the Solution Structure of Nonlinear Hill's Equation I,Global Results

The solution structure in W^2,p(?) × ? of nonlinear Hill's equation is discussed in full detail. In a recent article, the existence of unbounded solution components was shown for values of the branching parameter in the gaps of the continuous spectrum of the linearized problem. This result is the starting point of further investigations concerning the existence region of the corresponding solution components. In particular, new phenomena such as asymptotic bifurcation from infinity at a specific parameter value inside of each gap of the spectrum can be shown, if the nonlinearity satisfies a growth condition. The main assumption is the concentration of the nonlinearity to a compact interval which allows the reduction to an equivalent nonlinear Sturm - Liouville problem with parameter dependent boundary conditions, if the parameter does not belong to the continuous spectrum. Extending this problem to all real parameter values makes it possible to get information about the existence region of the solution components with the help of a priori bounds for solutions of the Sturm-Liouville problem.     

Second-order general perturbed sweeping process differential inclusion

The paper presents a study of perturbed sweeping process where the moving set depends on both the time and the state. This evolution problem is governed by second-order differential inclusions with an unbounded perturbation. Assuming that such set is prox-regular or subsmooth, we prove the existence of solutions even in the presence of a delay.     

Unboundedness in reverse convex and concave integer programming

In this paper we are concerned with the problem of unboundedness and existence of an optimal solution in reverse convex and concave integer optimization problems. In particular, we present necessary and sufficient conditions for existence of an upper bound for a convex objective function defined over the feasible region contained in ${\mathbb{Z}^n}$ . The conditions for boundedness are provided in a form of an implementable algorithm, showing that for the considered class of functions, the integer programming problem is unbounded if and only if the associated continuous problem is unbounded. We also address the problem of boundedness in the global optimization problem of maximizing a convex function over a set of integers contained in a convex and unbounded region. It is shown in the paper that in both types of integer programming problems, the objective function is either unbounded from above, or it attains its maximum at a feasible integer point.     

A Stochastic Tikhonov Theorem in Infinite Dimensions

The present paper studies the problem of singular perturbation in the infinite-dimensional framework and gives a Hilbert-space-valued stochastic version of the Tikhonov theorem. We consider a nonlinear system of Hilbert-space-valued equations for a "slow" and a "fast" variable; the system is strongly coupled and driven by linear unbounded operators generating a C₀-semigroup and independent cylindrical Brownian motions. Under well-established assumptions to guarantee the existence and uniqueness of mild solutions, we deduce the required stability of the system from a dissipativity condition on the drift of the fast variable. We avoid differentiability assumptions on the coefficients which would be unnatural in the infinite-dimensional framework.     

Parabolic Differential Equations with Unbounded Coefficients – A Generalization of the Parametrix Method

We consider uniformly parabolic differential equations with unbounded first- and zero-order coefficients. A fundamental solution is constructed based on the classical parametrix method of E. Levi. From this the existence and uniqueness of the corresponding Cauchy problem is derived. Our approach does not require differentiable coefficients, as is usually assumed in the unbounded case. It only requires Hölder continuous coefficients. In this respect, our new proof also extends known results. We briefly discuss applications which make essential use of this extension.     

Propagation of perturbations in a two-layer stratified fluid with an interface excited by moving sources

Propagation of small perturbations in a two-layer inviscid stratified fluid is studied. It is assumed that the higher density fluid occupies the lower unbounded half-space, while the lower density fluid occupies the upper unbounded half-space. The source of the excitation is a plane wave traveling along the interface of the fluids. An explicit analytical solution to the problem is constructed, and its existence and uniqueness are proved. The long-time wave pattern developing in the fluids is analyzed.