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We develop the method of vector-fields to further study Dispersive Wave Equations. Radial vector fields are used to get a-priori estimates such as the Morawetz estimate on solutions of Dispersive Wave Equations. A key to such estimates is…
In this paper, we employed linear algebra and functional analysis to determine necessary and sufficient conditions for oscillation-free and stable solutions to linear and nonlinear parabolic partial differential equations. We applied…
In this article, we present an efficient numerical model able to solve the vectorial nonlinear pulse propagation equation in circularly symmetric multimode waveguides. The algorithm takes advantage of the conservation of total angular…
This article presents the numerical eigensolver to find the resonant frequencies of 3-D closed cavity resonators filled with both electric and magnetic lossy, anisotropic media. By introducing a dummy variable with zero value in the 3-D…
Adaptive finite elements are the method of choice for accurate simulations of optical components. However as shown recently by Bienstman et al. many finite element mode solvers fail to compute the propagation constant's imaginary part of a…
We develop a high order accurate numerical method for solving the elastic wave equation in second-order form. We hybridize the computationally efficient Cartesian grid formulation of finite differences with geometrically flexible…
A widespread, intuitive and computationally inexpensive method to analyze light guidance through waveguide bends is by introducing an equivalent straight waveguide with refractive index profile modified to account for actual waveguide bend.…
Optical properties of hybrid plasmonic waveguides and of low-Q cavities, formed by waveguides of finite length are investigated numerically. These structures are of interest as building-blocks of plasmon lasers. We use a time-harmonic…
Large-scale simulations of the wave equation in electromagnetism, seismology, and acoustics, can be solved efficiently by finite difference methods. The accuracy of these numerical solutions usually depends on the minimization of…
When the electromagnetic wave is incident on the periodic structures, in addition to the scattering field, some guided modes that are traveling in the periodic medium could be generated. In the present paper, we study the calculation of…
We present our simulation tool JCMmode for calculating propagating modes of an optical waveguide. As ansatz functions we use higher order, vectorial elements (Nedelec elements, edge elements). Further we construct transparent boundary…
In this paper we study the problem of wave propagation in a 3-D optical fiber. The goal is to obtain a solution for the time-harmonic field caused by a source in a cylindrically symmetric waveguide. The geometry of the problem,…
From a mathematical perspective, the extraordinary properties of metamaterials are often reflected in the coefficients of the governing partial differential equations (PDEs). These coefficients may fall outside the assumptions of classical…
The diffusive viscous wave equation describes wave propagation in diffusive and viscous media. Examples include seismic waves traveling through the Earth's crust, taking into account of both the elastic properties of rocks and the…
Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate framework for solving the wave equation in second order form. That said, the standard…
Generalized solitary waves with exponentially small non-decaying far field oscillations have been studied in a range of singularly-perturbed differential equations, including higher-order Korteweg-de Vries (KdV) equations. Many of these…
Rigorous computer simulations of propagating electromagnetic fields have become an important tool for optical metrology and design of nanostructured optical components. A vectorial finite element method (FEM) is a good choice for an…
The aim of this lecture is to consider a concrete problem, namely the identification of situations of invisibility in waveguides, to present techniques and tools that may be useful in various fields of applied mathematics. To be more…
Fluidic Shaping is a novel method for fabrication of optical components based on the equilibrium state of liquid volumes in neutral buoyancy, subjected to geometrical constraints. The underlying physics of this method is described by a…
In this paper, we consider a variational formulation for the Dirichlet problem of the wave equation with zero boundary and initial conditions, where we use integration by parts in space and time. To prove unique solvability in a subspace of…