We survey recent wellposedness and illposedness results for the Cauchy problem for nonlinear Schrödinger equations in inhomogeneous media such as Riemannian manifolds or domains of the Euclidean space, trying to emphasize the influence of the geometry. The main tools are multilinear Strichartz estimates for the Schrödinger group. Mathematics Subject Classification (2000). Primary 35Q55; Seconda...

The form of the phenomenological stress-energy-momentum tensor for the electromagnetic field in a class of inhomogeneous, anisotropic magneto-electric media is calculated from first principles, leading to a coherent understanding of the phenomenological stresses and energy-momentum exchanges induced by electromagnetic interactions with such matter in terms of a fully relativistic covariant vari...

This paper deals with the scattering problem by an inhomogeneous object embedded in a chiral medium. A combined Hilbert space and integral equation method is employed to prove the existence and uniqueness of the solutions to the scattering problem. It is shown that the classical Silver-M uller radiation condition remains valid in chiral media.

The Cell Method, a new efficient numerical method suitable for working with periodic structures having anisotropic inhomogeneous media with curved shapes, is proposed in order to calculate the band gap of 2D photonic crystals for in-plane propagation of TM and TE waves. Moreover some numerical comparisons with other numerical methods will be provided.

In this thesis, we introduce a new, fast, high-order method for scattering by inhomogeneous media in three dimensions. As in previously existing methods, the low (O(N logN)) complexity of our integral equation method is obtained through extensive use of the fast Fourier transform (FFT) in evaluating the required convolutions. Unlike previous FFT-based methods, however, this method yields high-o...

The influence of topological defects on phase synchronization and phase coherence in two-dimensional arrays of locally coupled, nonidentical, chaotic oscillators is investigated. The motion of topological defects leads to a breakdown of phase synchronization in the vicinities of the defects; however, the system is much more phase coherent as long as the coupling between the oscillators is stron...

In the reconstruction process of unknown multiple scattering objects in inverse medium scattering problems, the first important step is to effectively locate some approximate domains that contain all inhomogeneous media. Without such an effective step, one may have to take a computational domain of the size that is much larger than the actual sizes of all scattering objects, thus resulting in a...

This thesis deals with the propagation of optical waves in Kerr nonlinear media, with emphasis on optical beam propagation beyond the slowly varying envelope approximation (SVEA). The first topic in this thesis is the propagation of a continuous-wave in one-dimensional (1D) nonlinear grating structures. For this purpose we develop a numerical scheme based on a variational method. It directly im...

In this paper, we present linear theory and nonlinear simulations to study the self-similar growth and shrinkage of a precipitate in a 2D elastic media. This work is motivated by a series of studies by Li et al., where the existence and morphological stability of self-similarly evolving crystals were demonstrated in a diffusional field. Here, we extend the theory and simulations to account for ...

The problem studied in this paper is ultrasound image reconstruction from frequency-domain measurements of the scattered field from an object with contrast in attenuation and sound speed. The case where the object has uniform but unknown contrast in these properties relative to the background is considered. Background clutter is taken into account in a physically realistic manner by considering...