A linear sampling method for inverse problems of diffraction gratings of mixed type

This paper is concerned with the direct and inverse problem of scattering of a time‐harmonic wave by a Lipschitz diffraction grating of mixed type. The scattering problem is modeled by the mixed boundary value problem for the Helmholtz equation in the unbounded half‐plane domain above a periodic Lipschitz surface on which a mixed Dirichlet and impedance boundary condition is imposed. We first establish the well‐posedness of the direct problem, employing the variational method, and then extend Isakov's method to prove uniqueness in determining the Lipschitz diffraction grating profile by using point sources lying above the structure. Finally, we develop a periodic version of the linear sampling method to reconstruct the diffraction grating. In this case, the far field equation defined on the unit circle is replaced by a near field equation defined on a line above the surface, which is a linear integral equation of the first kind. Numerical results are also presented to illustrate the efficiency of the method in the case when the height of the unknown grating profile is not very large and the noise level of the near field measurements is not very high. Copyright © 2012 John Wiley & Sons, Ltd.     

Inverse diffraction by a doubly periodic structure

Let us consider the scattering of electromagnetic waves by a doubly periodic structure. Above the structure, the medium is assumed to be homogeneous with a fixed real dielectric coefficient. The medium is a perfect conductor below the structure. For a given incident plane wave, the tangential electric field is measured away from the structure. An inverse problem arises: To what extent can one determine the location of the periodic structure that separates the dielectric medium from the conductor? In this paper, results on uniqueness and stability are established for the inverse problem. A crucial step in our proof is to obtain a lower bound for the first eigenvalue of an eigenvalue problem.     

On Doubly Periodic Standing Wave Solutions of the Coupled Higgs Field Equation

In this paper, we derive a class of doubly periodic standing wave solutions for a coupled Higgs field equation by employing the Hirota bilinear method and theta function identities. Such solutions are expressed in terms of theta functions with variable separation form. Moreover, it is shown that these solutions can be converted into Jacobi elliptic function representations, and their long‐wave limit yields collision of dark solitons. In comparing with known solutions of the canonical evolution equation, three new aspects will be developed in this paper. First, the periods in the spatial and temporal directions, measured in terms of the theta function parameters τ and τ₁, are independent of each other, quite unlike most similar solutions found earlier in the literature. Second, the doubly periodic wave solutions possess two families of the local maxima, whose locations and types are then examined in detail. Third, we obtain new doubly periodic standing wave solutions for the Davey–Stewartson equation through its similarity transformation to the coupled Higgs field equation.     

Direct and inverse problems for electromagnetic scattering by a doubly periodic structure with a partially coated dielectric

Consider the problem of scattering of electromagnetic waves by a doubly periodic Lipschitz structure. The medium above the structure is assumed to be homogenous and lossless with a positive dielectric coefficient. Below the structure there is a perfect conductor with a partially coated dielectric boundary. We first establish the well‐posedness of the direct problem in a proper function space and then obtain a uniqueness result for the inverse problem by extending Isakov's method. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamics of a class of ODEs more general than almost periodic

A temporally global solution, if it exists, of a nonautonomous ordinary differential equation need not be periodic, almost periodic or almost automorphic when the forcing term is periodic, almost periodic or almost automorphic, respectively. An alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here. Specifically, the class of functions consists of uniformly continuous functions, defined on the real line and taking values in a Banach space, which have pre-compact ranges. Besides periodic and almost periodic functions, this class also includes many nonrecurrent functions. Assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts, the existence of a special bounded entire solution, as well the existence of stable and unstable manifolds of this solution are established. Moreover, it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation. In the stable case it is shown that this special solution is the pullback attractor of the system. A class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a C₀-semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations.     

Sur un problème inverse pour les équations de Maxwell dans un réseau doublement périodique

Consider an electromagnetic plane wave incident on a doubly periodic structure in ø³. We prove that the knowledge of the tangential component of the electric field on an elementary section determines the electric permittivity of the grating.     

Difference Equation for Functions Analytic Outside Two Squares

We consider a totality of two squares built on primitive periods 1 and i and “sufficiently close to each other“. In a vicinity of this set we investigate four-element difference equation with constant coefficients, whose linear shifts are generating transforms of the corresponding doubly periodic group and the inverse transforms. We seek a solution in a class of functions, which are analytic outside this set and vanish at infinity. The equation is applicable to the moments problem for entire functions of exponential type.     

Meromorphic solutions of a Riccati differential equation with a doubly periodic coefficient

We treat a Riccati differential equation w^′+w²+p(z)=0, where p(z) is a nonconstant doubly periodic meromorphic function. Under certain assumptions, every solution is meromorphic in the whole complex plane. We show that the growth order of it is equal to 2, and examine the frequency of α-points and poles. Furthermore, the number of doubly periodic solutions is discussed.     

Existence of doubly periodic vortices in a generalized Chern–Simons model

We establish an existence theorem for the doubly periodic vortices in a generalized self-dual Chern–Simons model. We show that there exists a critical value of the coupling parameter such that there exist self-dual doubly periodic vortex solutions for the generalized self-dual Chern–Simons equation if and only if the coupling parameter is less than or equal to the value. The energy, magnetic flux, and electric charge associated to the field configurations are all specifically quantized. By the solutions obtained for this generalized self-dual Chern–Simons equation we can also construct doubly periodic vortex solutions to a related generalized self-dual Abelian Higgs equation.     

An existence principle for a periodic solution of a differential equation in Banach space

The equation d²x/dt²=Ax +f(t, x) is considered in a Banach space E, where A is a fixed unbounded linear operator, andf(t, x) is a nonlinear operator which is periodic in t and satisfies a Lipschitz condition with respect to x E. Existence conditions have been obtained for a well defined generalized periodic solution of this equation, and also when this solution coincides with the true solution. Similar results have been obtained for the first order equation.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 105–112, July, 1968.     

Difference Equations for Functions Analytic Beyond Several Squares

We consider some set of squares constructed for the primitive periods 1 and i and sufficiently distant from each other. In a neighborhood of this set we study a four-element difference equation with constant coefficients whose linear stifts are generators of the corresponding doubly periodic group and their inverses. A solution is sought in the class of functions analytic beyond this set and vanishing at infinity. We show that the solvability of the problem depends essentially not only on the choice of the coefficients but also on the disposition of the squares.     

Relative asymptotic behavior of pseudoholomorphic half‐cylinders

In this article we study the asymptotic behavior of pseudoholomorphic half‐cylinders that converge exponentially to a periodic orbit of a vector field defined by a framed stable Hamiltonian structure. Such maps are of central interest in symplectic field theory and its variants (symplectic Floer homology, contact homology, and embedded contact homology). We prove a precise formula for the asymptotic behavior of the “difference” of two such maps, generalizing results from [6, 7, 12, 15]. Using this result with a technique from [14], we then show that a finite collection of pseudoholomorphic half‐cylinders asymptotic to coverings of a single periodic orbit is smoothly equivalent to solutions to a linear equation. © 2007 Wiley Periodicals, Inc.     

Periodic orbits and unstable manifolds for ordinary differential equations

Consider the unstable manifold of a hyperbolic periodic orbit of an ordinary differential equation under C¹ perturbations of the vector field and under approximation by a one-step numerical method, which is at least first order. Trajectories bounded backwards in time near the periodic orbit perturb Hausdorff continuously. This result as applied to numerical perturbations improves on Alouges-Debussche [1], who give only continuity of the unstable maniford, and on Beyn [3], who gives continuity of trajectories only when the periodic orbit is unstable. As a corollary, we find that attractors perturb Hausdorff continuously when the attractor equals a union of locally continuous unstable manifolds of invariant sets     

A Linear PDE Approach to the Bellman Equation of Ergodic Control with Periodic Structure

Abstract. In this paper we give a new proof of the existence result of Bensoussan [1, Theorem II-6.1] for the Bellman equation of ergodic control with periodic structure. This Bellman equation is a nonlinear PDE, and he constructed its solution by using the solution of a nonlinear PDE. On the contrary, our key idea is to solve two linear PDEs. Hence, we propose a linear PDE approach to this Bellman equation.     

Multiple vortices for a self-dual CP(1) Maxwell-Chern-Simons model

We prove the existence of at least two doubly periodic vortex solutions for a self-dual CP(1) Maxwell-Chern-Simons model. To this end we analyze a system of two elliptic equations with exponential nonlinearities. Such a system is shown to be equivalent to a fourth-order elliptic equation admitting a variational structure. Tonia Ricciardi: Partially supported by the MIUR National Project Variational Methods and Nonlinear Differential Equations     

电报方程双周期解的极大值原理与强正性估计及应用

本文讨论非线性电报方程u_(tt)-u_(xx)+cu_t=F(t,x,u),(t,x)∈R~2时空双2π周期解的存在性。改进了Ortega与Robles-Perez关于线性电报方程双周期解的极大值原理,应用新获得的极大值原理,推广了相应的上下解定理,并且加强了极大值原理的结论,建立了线性方程解的强正性估计,利用这个强正性估计及锥上的不动点定理获得了超线性电报方程及奇异电报方程正双周期解的存在性。     

A Linear PDE Approach to the Bellman Equation of Ergodic Control with Periodic Structure

   Abstract. In this paper we give a new proof of the existence result of Bensoussan [1, Theorem II-6.1] for the Bellman equation of ergodic control with periodic structure. This Bellman equation is a nonlinear PDE, and he constructed its solution by using the solution of a nonlinear PDE. On the contrary, our key idea is to solve two linear PDEs. Hence, we propose a linear PDE approach to this Bellman equation.     

Squeezed states and their applications to quantum evolution

In this paper, we consider quantum multidimensional problems solvable by using the second quantization method. A multidimensional generalization of the Bogolyubov factorization formula, which is an important particular case of the Campbell-Baker-Hausdorff formula, is established. The inner product of multidimensional squeezed states is calculated explicitly; this relationship justifies a general construction of orthonormal systems generated by linear combinations of squeezed states. A correctly defined path integral representation is derived for solutions of the Cauchy problem for the Schrödinger equation describing the dynamics of a charged particle in the superposition of orthogonal constant (E,H)-fields and a periodic electric field. We show that the evolution of squeezed states runs over compact one-dimensional matrix-valued orbits of squeezed components of the solution, and the evolution of coherent shifts is a random Markov jump process which depends on the periodic component of the potential.     

Homogenization of the 3D Maxwell system near resonances and artificial magnetism

It is now well known that the homogenization of a periodic array of parallel dielectric fibers with suitably scaled high permittivity can lead to a possibly negative frequency dependent effective permeability. However this result based on a two-dimensional micro resonator problem on the section of the fibers holds merely in the case of polarized magnetic fields, reducing thus its applications to infinite cylindrical obstacles. In this Note we propose a full 3D extension of previous asymptotic analysis based on a new averaging method for the magnetic field. We evidence a vectorial spectral problem on the periodic cell which accounts for micro-resonance effects and leads to a 3D negative effective permeability tensor. This suggests that periodic bulk dielectric inclusions could be an efficient alternative to the very popular metallic split-ring structure proposed by Pendry. To cite this article: G. Bouchitté et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).