Asymptotic Analysis - Volume 103, issue 3

Authors: Delourme, Bérangère | Fliss, Sonia | Joly, Patrick | Vasilevskaya, Elizaveta

Article Type: Research Article

Abstract: The present paper deals with the wave propagation in a particular two dimensional structure, obtained from a localized perturbation of a reference periodic medium. This reference medium is a ladder like domain, namely a thin periodic structure (the thickness being characterized by a small parameter ε > 0 ) whose limit (as ε tends to 0) is a periodic graph. The localized perturbation consists in changing the geometry of the reference medium by modifying the thickness of one rung of the ladder. Considering the scalar Helmholtz equation with Neumann boundary conditions in this domain, we wonder whether …such a geometrical perturbation is able to produce localized eigenmodes. To address this question, we use a standard approach of asymptotic analysis that consists of three main steps. We first find the formal limit of the eigenvalue problem as the ε tends to 0. In the present case, it corresponds to an eigenvalue problem for a second order differential operator defined along the periodic graph. Then, we proceed to an explicit calculation of the spectrum of the limit operator. Finally, we prove that the spectrum of the initial operator is close to the spectrum of the limit operator. In particular, we prove the existence of localized modes provided that the geometrical perturbation consists in diminishing the width of one rung of the periodic thin structure. Moreover, in that case, it is possible to create as many eigenvalues as one wants, provided that ε is small enough. Numerical experiments illustrate the theoretical results. Show more

Keywords: Spectral theory, periodic media, quantum graphs, trapped modes

DOI: 10.3233/ASY-171422

Citation: Asymptotic Analysis, vol. 103, no. 3, pp. 103-134, 2017

Authors: Kachmar, Ayman | Nasrallah, Marwa

Article Type: Research Article

Abstract: The energy of a type II superconductor placed in a strong non-uniform, smooth and signed magnetic field is displayed via a universal reference function defined by means of a simplified two dimensional Ginzburg–Landau functional. We study the asymptotic behavior of this functional in a specific asymptotic regime, thereby linking it to a one dimensional functional, using methods developed by Almog–Helffer and Fournais–Helffer devoted to the analysis of surface superconductivity in the presence of a uniform magnetic field. As a result, we obtain an asymptotic formula reminiscent of the one for the surface superconductivity regime, where the zero set of the …magnetic field plays the role of the superconductor’s surface. Show more

Keywords: Superconductivity, Ginzburg–Landau functional, ground state energy, vanishing magnetic field, Montgomery operator

DOI: 10.3233/ASY-171424

Citation: Asymptotic Analysis, vol. 103, no. 3, pp. 135-163, 2017