Pseudo-Differential Operators in a Wave Diffraction Problem with Impedance Conditions

Mathematics Subject Classification: 35J05, 35J25, 35C15, 47H50, 47G30

Stability and Instability of Solitary Wave Solutions of a Nonlinear Dispersive System of Benjamin-Bona-Mahony Type

2000 Mathematics Subject Classification: 35B35, 35B40, 35Q35, 76B25, 76E30.

Singular Solutions of Protter’s Problem for a Class of 3-D Hyperbolic Equations

Недю Иванов Попиванов, Алексей Йорданов Николов - През 1952 г. М. Протър формулира нови гранични задачи за вълновото уравнение, които са тримерни аналози на задачите на Дарбу в равнината. Задачите са разгледани в тримерна област, ограничена от две характеристични конуса и равнина. Сега, след като са минали повече от 50 години, е добре известно, че за безброй гладки функции в дясната страна на уравнението тези задачи нямат класически решения, а обобщеното решение има силна степенна особеност във върха на характеристичния конус, която е изолирана и не се разпространява по конуса. Тук ние разглеждаме трета гранична задача за вълновото уравнение с младши членове и дясна страна във формата на тригонометричен полином. Дадена е по-нова от досега известната априорна оценка за максимално възможната особеност на решенията на тази задача. Оказва се, че при по-общото уравнение с младши членове възможната сингулярност е от същия ред като при чисто вълновото уравнение.

Impact of inter-modal four-wave mixing on the performance of mode- and wavelength-division-multiplexing systems

In this paper, we investigate the impact of inter-modal four-wave mixing on mode- and wavelength-division-multiplexing systems. A set of coupled nonlinear Schrödinger equations, including linear mode coupling, is derived allowing to isolate the inter-modal four-wave mixing terms. The efficiency of inter-modal four-wave mixing between degenerate LP modes is found to be significantly higher than the intra-modal four-wave mixing efficiency. However, it is shown that the inter-modal four-wave mixing efficiency between degenerate modes is significantly reduced by the linear mode coupling.

A Simple Example of Localized Parametric Resonance for the Wave Equation

2000 Mathematics Subject Classification: 35L05, 35P25, 47A40.

Dunkl-Schrödinger Equations with and without Quadratic Potentials

2000 Mathematics Subject Classification: Primary 42A38. Secondary 42B10.

One-dimensional optical wave turbulence:experiment and theory

We present a review of the latest developments in one-dimensional (1D) optical wave turbulence (OWT). Based on an original experimental setup that allows for the implementation of 1D OWT, we are able to show that an inverse cascade occurs through the spontaneous evolution of the nonlinear field up to the point when modulational instability leads to soliton formation. After solitons are formed, further interaction of the solitons among themselves and with incoherent waves leads to a final condensate state dominated by a single strong soliton. Motivated by the observations, we develop a theoretical description, showing that the inverse cascade develops through six-wave interaction, and that this is the basic mechanism of nonlinear wave coupling for 1D OWT. We describe theory, numerics and experimental observations while trying to incorporate all the different aspects into a consistent context. The experimental system is described by two coupled nonlinear equations, which we explore within two wave limits allowing for the expression of the evolution of the complex amplitude in a single dynamical equation. The long-wave limit corresponds to waves with wave numbers smaller than the electrical coherence length of the liquid crystal, and the opposite limit, when wave numbers are larger. We show that both of these systems are of a dual cascade type, analogous to two-dimensional (2D) turbulence, which can be described by wave turbulence (WT) theory, and conclude that the cascades are induced by a six-wave resonant interaction process. WT theory predicts several stationary solutions (non-equilibrium and thermodynamic) to both the long- and short-wave systems, and we investigate the necessary conditions required for their realization. Interestingly, the long-wave system is close to the integrable 1D nonlinear Schrödinger equation (NLSE) (which contains exact nonlinear soliton solutions), and as a result during the inverse cascade, nonlinearity of the system at low wave numbers becomes strong. Subsequently, due to the focusing nature of the nonlinearity, this leads to modulational instability (MI) of the condensate and the formation of solitons. Finally, with the aid of the probability density function (PDF) description of WT theory, we explain the coexistence and mutual interactions between solitons and the weakly nonlinear random wave background in the form of a wave turbulence life cycle (WTLC).

Time-domain simulation of wave-induced ship motions by a Rankine panel method

In this thesis, a numerical program has been developed to simulate the wave-induced ship motions in the time domain. Wave-body interactions have been studied for various ships and floating bodies through forced motion and free motion simulations in a wide range of wave frequencies. A three-dimensional Rankine panel method is applied to solve the boundary value problem for the wave-body interactions. The velocity potentials and normal velocities on the boundaries are obtained in the time domain by solving the mixed boundary integral equations in relation to the source and dipole distributions. The hydrodynamic forces are calculated by the integration of the instantaneous hydrodynamic pressures over the body surface. The equations of ship motion are solved simultaneously with the boundary value problem for each time step. The wave elevation is computed by applying the linear free surface conditions. A numerical damping zone is adopted to absorb the outgoing waves in order to satisfy the radiation condition for the truncated free surface. A numerical filter is applied on the free surface for the smoothing of the wave elevation. Good convergence has been reached for both forced motion simulations and free motion simulations. The computed added-mass and damping coefficients, wave exciting forces, and motion responses for ships and floating bodies are in good agreement with the numerical results from other programs and experimental data.

Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation

We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an observer on the boundary of the manifold when the manifold is set into motion through boundary vibrations. We study the Dirichlet to Neumann operator when vibrations are imposed and data recorded on disjoint sets, a useful setting for applications. We prove that this operator determines the Dirichlet to Neumann operator where sources and observations are on the same set, provided a spectral condition on the Laplace-Beltrami operator for the manifold is satisfied. We prove this by providing an implementable procedure for determining a portion of the Riemannian manifold near the area where sources are applied. Drawing on established results, an immediate corollary is that a compact Riemannian manifold can be reconstructed from the Dirichlet to Neumann operator where sources and observations are on disjoint sets.

Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation

We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an observer on the boundary of the manifold when the manifold is set into motion through boundary vibrations. We study the Dirichlet to Neumann operator when vibrations are imposed and data recorded on disjoint sets, a useful setting for applications. We prove that this operator determines the Dirichlet to Neumann operator where sources and observations are on the same set, provided a spectral condition on the Laplace-Beltrami operator for the manifold is satisfied. We prove this by providing an implementable procedure for determining a portion of the Riemannian manifold near the area where sources are applied. Drawing on established results, an immediate corollary is that a compact Riemannian manifold can be reconstructed from the Dirichlet to Neumann operator where sources and observations are on disjoint sets.

Projected equations of motion approach to hybrid quantum/classical dynamics in dielectric-metal composites

We introduce a hybrid method for dielectric-metal composites that describes the dynamics of the metallic system classically whilst retaining a quantum description of the dielectric. The time-dependent dipole moment of the classical system is mimicked by the introduction of projected equations of motion (PEOM) and the coupling between the two systems is achieved through an effective dipole-dipole interaction. To benchmark this method, we model a test system (semiconducting quantum dot-metal nanoparticle hybrid). We begin by examining the energy absorption rate, showing agreement between the PEOM method and the analytical rotating wave approximation (RWA) solution. We then investigate population inversion and show that the PEOM method provides an accurate model for the interaction under ultrashort pulse excitation where the traditional RWA breaks down.

Numerical Simulation Of Ultrasonic Wave Propagation In Elastically Anisotropic Media

The ultrasonic non-destructive testing of components may encounter considerable difficulties to interpret some inspections results mainly in anisotropic crystalline structures. A numerical method for the simulation of elastic wave propagation in homogeneous elastically anisotropic media, based on the general finite element approach, is used to help this interpretation. The successful modeling of elastic field associated with NDE is based on the generation of a realistic pulsed ultrasonic wave, which is launched from a piezoelectric transducer into the material under inspection. The values of elastic constants are great interest information that provide the application of equations analytical models, until small and medium complexity problems through programs of numerical analysis as finite elements and/or boundary elements. The aim of this work is the comparison between the results of numerical solution of an ultrasonic wave, which is obtained from transient excitation pulse that can be specified by either force or displacement variation across the aperture of the transducer, and the results obtained from a experiment that was realized in an aluminum block in the IEN Ultrasonic Laboratory. The wave propagation can be simulated using all the characteristics of the material used in the experiment evaluation associated to boundary conditions and from these results, the comparison can be made.

Wavelength converters based on four-wave mixing in semiconductor optical amplifiers for advanced modulation formats

Les convertisseurs de longueur d’onde sont essentiels pour la réalisation de réseaux de communications optiques à routage en longueur d’onde. Dans la littérature, les convertisseurs de longueur d’onde basés sur le mélange à quatre ondes dans les amplificateurs optiques à semi-conducteur constituent une solution extrêmement intéressante, et ce, en raison de leurs nombreuses caractéristiques nécessaires à l’implémentation de tels réseaux de communications. Avec l’émergence des systèmes commerciaux de détection cohérente, ainsi qu’avec les récentes avancées dans le domaine du traitement de signal numérique, il est impératif d’évaluer la performance des convertisseurs de longueur d’onde, et ce, dans le contexte des formats de modulation avancés. Les objectifs de cette thèse sont : 1) d’étudier la faisabilité des convertisseurs de longueur d’onde basés sur le mélange à quatre ondes dans les amplificateurs optiques à semi-conducteur pour les formats de modulation avancés et 2) de proposer une technique basée sur le traitement de signal numérique afin d’améliorer leur performance. En premier lieu, une étude expérimentale de la conversion de longueur d’onde de formats de modulation d’amplitude en quadrature (quadrature amplitude modulation - QAM) est réalisée. En particulier, la conversion de longueur d’onde de signaux 16-QAM à 16 Gbaud et 64-QAM à 5 Gbaud dans un amplificateur optique à semi-conducteur commercial est réalisée sur toute la bande C. Les résultats démontrent qu’en raison des distorsions non-linéaires induites sur le signal converti, le point d’opération optimal du convertisseur de longueur d’onde est différent de celui obtenu lors de la conversion de longueur d’onde de formats de modulation en intensité. En effet, dans le contexte des formats de modulation avancés, c’est le compromis entre la puissance du signal converti et les non-linéarités induites qui détermine le point d’opération optimal du convertisseur de longueur d’onde. Les récepteurs cohérents permettent l’utilisation de techniques de traitement de signal numérique afin de compenser la détérioration du signal transmis suite à sa détection. Afin de mettre à profit les nouvelles possibilités offertes par le traitement de signal numérique, une technique numérique de post-compensation des distorsions induites sur le signal converti, basée sur une analyse petit-signal des équations gouvernant la dynamique du gain à l’intérieur des amplificateurs optiques à semi-conducteur, est développée. L’efficacité de cette technique est démontrée à l’aide de simulations numériques et de mesures expérimentales de conversion de longueur d’onde de signaux 16-QAM à 10 Gbaud et 64-QAM à 5 Gbaud. Cette méthode permet d’améliorer de façon significative les performances du convertisseur de longueur d’onde, et ce, principalement pour les formats de modulation avancés d’ordre supérieur tel que 64-QAM. Finalement, une étude expérimentale exhaustive de la technique de post-compensation des distorsions induites sur le signal converti est effectuée pour des signaux 64-QAM. Les résultats démontrent que, même en présence d’un signal à bruité à l’entrée du convertisseur de longueur d’onde, la technique proposée améliore toujours la qualité du signal reçu. De plus, une étude du point d’opération optimal du convertisseur de longueur d’onde est effectuée et démontre que celui-ci varie en fonction des pertes optiques suivant la conversion de longueur d’onde. Dans un réseau de communication optique à routage en longueur d’onde, le signal est susceptible de passer par plusieurs étages de conversion de longueur d’onde. Pour cette raison, l’efficacité de la technique de post-compensation est démontrée, et ce pour la première fois dans la littérature, pour deux étages successifs de conversion de longueur d’onde de signaux 64-QAM à 5 Gbaud. Les résultats de cette thèse montrent que les convertisseurs de longueur d’ondes basés sur le mélange à quatre ondes dans les amplificateurs optiques à semi-conducteur, utilisés en conjonction avec des techniques de traitement de signal numérique, constituent une technologie extrêmement prometteuse pour les réseaux de communications optiques modernes à routage en longueur d’onde.

Evans functions for integral neural field equations with Heaviside firing rate function

In this paper we show how to construct the Evans function for traveling wave solutions of integral neural field equations when the firing rate function is a Heaviside. This allows a discussion of wave stability and bifurcation as a function of system parameters, including the speed and strength of synaptic coupling and the speed of axonal signals. The theory is illustrated with the construction and stability analysis of front solutions to a scalar neural field model and a limiting case is shown to recover recent results of L. Zhang [On stability of traveling wave solutions in synaptically coupled neuronal networks, Differential and Integral Equations, 16, (2003), pp.513-536.]. Traveling fronts and pulses are considered in more general models possessing either a linear or piecewise constant recovery variable. We establish the stability of coexisting traveling fronts beyond a front bifurcation and consider parameter regimes that support two stable traveling fronts of different speed. Such fronts may be connected and depending on their relative speed the resulting region of activity can widen or contract. The conditions for the contracting case to lead to a pulse solution are established. The stability of pulses is obtained for a variety of examples, in each case confirming a previously conjectured stability result. Finally we show how this theory may be used to describe the dynamic instability of a standing pulse that arises in a model with slow recovery. Numerical simulations show that such an instability can lead to the shedding of a pair of traveling pulses.

Modeling electrocortical activity through improved local approximations of integral neural field equations

Neural field models of firing rate activity typically take the form of integral equations with space-dependent axonal delays. Under natural assumptions on the synaptic connectivity we show how one can derive an equivalent partial differential equation (PDE) model that properly treats the axonal delay terms of the integral formulation. Our analysis avoids the so-called long-wavelength approximation that has previously been used to formulate PDE models for neural activity in two spatial dimensions. Direct numerical simulations of this PDE model show instabilities of the homogeneous steady state that are in full agreement with a Turing instability analysis of the original integral model. We discuss the benefits of such a local model and its usefulness in modeling electrocortical activity. In particular we are able to treat "patchy'" connections, whereby a homogeneous and isotropic system is modulated in a spatially periodic fashion. In this case the emergence of a "lattice-directed" traveling wave predicted by a linear instability analysis is confirmed by the numerical simulation of an appropriate set of coupled PDEs. Article published and (c) American Physical Society 2007