Coalescence of electromagnetic travelling waves in a saturated ferrite

We investigate how dissipation and nonlinearity affect an electromagnetic perturbation propagating into a saturated ferromagnet in the presence of an external magnetic field. We study the problem in (1+1) and (2+1) dimensions. It is found that at lowest order of the perturbation theory, the Burgers\u27 equation in (1+1) dimensions governs such dynamics. In (2+1) dimensions we show that the phenomena obeys a nonlinear evolution equation (non-integrable) of Burgers type. We give exact solutions which describe in (1+1) dimensions the propagation of a travelling electromagnetic wave and the coalescence of N travelling fronts and in (2+1) dimensions the propagation of a nearly one-dimensional travelling front. We establish, in terms of the physical parameters of the system, whether breaking or diffusion of the initial perturbation dominates

Transverse stability of short line-solitons in ferromagnetic media

In this work the propagation of nonlinear electromagnetic short waves in a ferromagnetic medium is discussed. It is shown that such waves propagate perpendicular to the magnetization density. The evolution of the wave under the influence of perturbations in one transverse dimension is considered; the asymptotic model equation governing the dynamics is a (2 + 1) generalization of the well-known sine-Gordon model. We exhibit the line-soliton solution and study its transverse stability. A numerical study of the model corroborates our analytical predictions

Two-dimensional electromagnetic solitons in the short-wave approximation

Date du colloque&nbsp;: 08/2010</p

Two-dimensional electromagnetic solitons in a perpendicularly magnetized ferromagnetic slab

We consider the nonlinear propagation of bulk polaritons in a magnetically saturated ferromagnetic thick film, the applied magnetic field being perpendicular to the film plane. A (2+1)-dimensional asymptotic model equation generalizing the sine-Gordon one is derived. Line-soliton solutions are exhibited, their stability condition is derived. When unstable, line solitons decay into stable two-dimensional lumps, which are studied by means of variational analysis

An integrable evolution equation for surface waves in deep water

In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative method, an asymptotic model for small wave steepness ratio is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical results is performed

Focusing and defocusing of electromagnetic waves in a ferromagnet

We show that the modulation of an elemmagnetic wave in a ferromagnet in the presence of an external magnetic field is govemed by the nonlinear Schradinger equation. We characterize the existence of an oscillatory instability of the Benjamin-Feir type for electromagnetic propagalion in a saturated ferrite. Depending on the physical parameters of the system, we establish the regions in which focusing or defocusing of the initial carrier envelope occurs. We show that one of the two plane wves, cicularly polarized, allowed to pmpagate into a ferrite, is bunching into solitons and the other is modulated by a dark soliton

Patents, market structure and welfare: A theoretical investigation into new dimensions of the patent system.

Chapter I: The relationship between inventor and the Patent Office is modelled as a 'patent regulation game' and it is shown that the conventional wisdom that the P.O. always maximizes welfare by playing the Stackelberg leader is incorrect. Other solution concepts are explored and it is found that, because of the patent life constraint, a reversal of roles may be beneficial. The result that social welfare can be maximized by the P.O. being a Stackelberg follower survives (albeit for a narrower range of values of the key parameters) even if the P.O.-leader is endowed with the additional instrument of a compulsory royalty rate. Chapter II: A new twist is added to the debate on the Schumpeterian competition hypothesis, by considering the structure of the final-product market as a policy Instrument, set by the Patent Office by manipulating patentability standards. It is found that for a vast range of demand functions and under constant returns to scale, a patentability standard that allows for more than one patent to be granted within a given product/process class is welfare superior to the monopoly-generating first-past-the-post current system. If patent life is beyond the P.O.'s control and/or there are increasing returns, no patentability standard is unambiguously preferable. Chapter III: When Research and Development are modelled as two analytically distinct stages, the choice between patentability standards (whether to grant patents to research prototypes or to fully-developed products) is shown to affect the allocation of resources between Research and Development. It is shown that under a single-patent regime, granting patents to research prototypes is unambiguously welfare-improving, whereas under a multiple-patent regime a change to patents being granted to fully-developed products and the attending increase in market uncertainty may raise welfare. Chapter IV: The economics of the 'integer constraint' is analysed and it is found that proper treatment of the indivisibility of firms may reverse the qualitative conclusions of interger-unconstrained models. As an example, a product quality oligopoly model is examined and it is shown that not only the Chamberlinian excess entry result does not apply but also that a free-entry oligopoly and a socially managed industry may produce goods of identical quality, irrespective of the values of cross-derivatives deemed crucial in the literature. Moreover, the integer constraint is shown to provide an explanation for a positive correlation between profitability and concentration in a Cournot oligopoly model with free entry

Theory of small aspect ratio waves in deep water

In the limit of small values of the aspect ratio parameter (or wave steepness) which measures the amplitude of a surface wave in units of its wave-length, a model equation is derived from the Euler system in infinite depth (deep water) without potential flow assumption. The resulting equation is shown to sustain periodic waves which on the one side tend to the proper linear limit at small amplitudes, on the other side possess a threshold amplitude where wave crest peaking is achieved. An explicit expression of the crest angle at wave breaking is found in terms of the wave velocity. By numerical simulations, stable soliton-like solutions (experiencing elastic interactions) propagate in a given velocities range on the edge of which they tend to the peakon solution.Comment: LaTex file, 16 pages, 4 figure

Benjamin-Feir-type instability in a saturated ferrite: Transition between focusing and defocusing regimes for polarized electromagnetic waves

We prove the existence of an oscillatory instability of the Benjamin-Feir type for electromagnetic propagation in a saturated ferrite. We do this by reducing the nonlinear equations governing the propagation of electromagnetic waves in such a medium to the nonlinear Schrödinger equation. We characterize regions where focusing or defocusing of the initial carrier envelope occurs in a function of three physical parameters: the phase velocity, the quotient between the external magnetic field and the magnetization of saturation, and a third one related to the angle between the direction of propagation of the carrier wave and the external magnetic field. We show that there exist points of transition between focusing and defocusing regimes for left elliptically polarized waves. No such point exists for right elliptically polarized waves. We show that all circularly polarized waves propagating parallel to the external magnetic field are stable (unstable) if they have negative (positive) helicity

Hydrothermal Surface-Wave Instability and the Kuramoto-Sivashinsky Equation

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrothermal surface-wave instabilities may give rise to disturbances governed by the Kuramoto-Sivashinsky equation. A possible connection to hot-wire experiments is also discussed.Comment: 11 pages, RevTex, no figure