Evolução top-hat hidrodinâmica em campos de velocidades peculiares primordiais

We investigate the cosmology of the vacuum energy decaying into cold dark matter according to thermodynamics description of Alcaniz & Lima. We apply this model to analyze the evolution of primordial density perturbations in the matter that gave rise to the first generation of structures bounded by gravity in the Universe, called Population III Objects. The analysis of the dynamics of those systems will involve the calculation of a differential equation system governing the evolution of perturbations to the case of two coupled fluids (dark matter and baryonic matter), modeled with a Top-Hat profile based in the perturbation of the hydrodynamics equations, an efficient analytical tool to study the properties of dark energy models such as the behavior of the linear growth factor and the linear growth index, physical quantities closely related to the fields of peculiar velocities at any time, for different models of dark energy. The properties and the dynamics of current Universe are analyzed through the exact analytical form of the linear growth factor of density fluctuations, taking into account the influence of several physical cooling mechanisms acting on the density fluctuations of the baryonic component of matter during the evolution of the clouds of matter, studied from the primordial hydrogen recombination. This study is naturally extended to more general models of dark energy with constant equation of state parameter in a flat Universe

Processos difusivos generalizados

We investigate several diffusion equations which extend the usual one by considering the presence of nonlinear terms or a memory effect on the diffusive term. We also considered a spatial time dependent diffusion coefficient. For these equations we have obtained a new classes of solutions and studied the connection of them with the anomalous diffusion process. We start by considering a nonlinear diffusion equation with a spatial time dependent diffusion coefficient. The solutions obtained for this case generalize the usual one and can be expressed in terms of the q-exponential and q-logarithm functions present in the generalized thermostatistics context (Tsallis formalism). After, a nonlinear external force is considered. For this case the solutions can be also expressed in terms of the q-exponential and q-logarithm functions. However, by a suitable choice of the nonlinear external force, we may have an exponential behavior, suggesting a connection with standard thermostatistics. This fact reveals that these solutions may present an anomalous relaxation process and then, reach an equilibrium state of the kind Boltzmann- Gibbs. Next, we investigate a nonmarkovian linear diffusion equation that presents a kernel leading to the anomalous diffusive process. Particularly, our first choice leads to both a the usual behavior and anomalous behavior obtained through a fractionalderivative equation. The results obtained, within this context, correspond to a change in the waiting-time distribution for jumps in the formalism of random walks. These modifications had direct influence in the solutions, that turned out to be expressed in terms of the Mittag-Leffler or H of Fox functions. In this way, the second moment associated to these distributions led to an anomalous spread of the distribution, in contrast to the usual situation where one finds a linear increase with time

Métodos numéricos para resolução de equações diferenciais ordinárias lineares baseados em interpolação por spline

In this work we have elaborated a spline-based method of solution of inicial value problems involving ordinary differential equations, with emphasis on linear equations. The method can be seen as an alternative for the traditional solvers such as Runge-Kutta, and avoids root calculations in the linear time invariant case. The method is then applied on a central problem of control theory, namely, the step response problem for linear EDOs with possibly varying coefficients, where root calculations do not apply. We have implemented an efficient algorithm which uses exclusively matrix-vector operations. The working interval (till the settling time) was determined through a calculation of the least stable mode using a modified power method. Several variants of the method have been compared by simulation. For general linear problems with fine grid, the proposed method compares favorably with the Euler method. In the time invariant case, where the alternative is root calculation, we have indications that the proposed method is competitive for equations of sifficiently high order.

Scattering and bound states of spinless particles in a mixed vector-scalar smooth step potential

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Hartman-Grobman decomposition when in presence of jumps in the relative degree of the dynamics in an energy harvesting device

This work deals with the nonlinear piezoelectric coupling in vibration-based energy harvesting, done by A. Triplett and D.D. Quinn in J. of Intelligent Material Syst. and Structures (2009). In that paper the first order nonlinear fundamental equation has a three dimensional state variable. Introducing both observable and control variables in such a way the controlled system became a SISO system, we can obtain as a corollary that for a particular choice of the observable variable it is possible to present an explicit functional relation between this variable one, and the variable representing the charge harvested. After-by observing that the structure in the Input-Output decomposition essentially changes depending on the relative degree changes, presenting bifurcation branches in its zero dynamics-we are able in to identify this type of bifurcation indicating its close relation with the Hartman - Grobman theorem telling about decomposition into stable and the unstable manifolds for hyperbolic points.

Zeros of Jacobi functions of second kind

The number of zeros in (- 1, 1) of the Jacobi function of second kind Q(n)((alpha, beta)) (x), alpha, beta > - 1, i.e. The second solution of the differential equation(1 - x(2))y (x) + (beta - alpha - (alpha + beta + 2)x)y' (x) + n(n + alpha + beta + 1)y(x) = 0,is determined for every n is an element of N and for all values of the parameters alpha > - 1 and beta > - 1. It turns out that this number depends essentially on alpha and beta as well as on the specific normalization of the function Q(n)((alpha, beta)) (x). Interlacing properties of the zeros are also obtained. As a consequence of the main result, we determine the number of zeros of Laguerre's and Hermite's functions of second kind. (c) 2005 Elsevier B.V. All rights reserved.

Stabilization of a (3+1)-dimensional soliton in a Kerr medium by a rapidly oscillating dispersion coefficient

Using the numerical solution of the nonlinear Schrodinger equation and a variational method it is shown that (3 + 1)-dimensional spatiotemporal optical solitons can be stabilized by a rapidly oscillating dispersion coefficient in a Kerr medium with cubic nonlinearity. This has immediate consequence in generating dispersion-managed robust optical soliton in communication as well as possible stabilized Bose-Einstein condensates in periodic optical-lattice potential via an effective-mass formulation. We also critically compare the present stabilization with that obtained by a rapid sinusoidal oscillation of the Kerr nonlinearity parameter.

Conformal Klein-Gordon equations and quasinormal modes

Using conformal coordinates associated with conformal relativity-associated with de Sitter spacetime homeomorphic projection into Minkowski spacetime-we obtain a conformal Klein-Gordon partial differential equation, which is intimately related to the production of quasi-normal modes (QNMs) oscillations, in the context of electromagnetic and/or gravitational perturbations around, e.g., black holes. While QNMs arise as the solution of a wave-like equation with a Poschl-Teller potential, here we deduce and analytically solve a conformal 'radial' d'Alembert-like equation, from which we derive QNMs formal solutions, in a proposed alternative to more completely describe QNMs. As a by-product we show that this 'radial' equation can be identified with a Schrodinger-like equation in which the potential is exactly the second Poschl-Teller potential, and it can shed some new light on the investigations concerning QNMs.

Atomic Bose-Einstein condensation with three-body interactions and collective excitations

The stability of a Bose-Einstein condensed state of trapped ultra-cold atoms is investigated under the assumption of an attractive two-body and a repulsive three-body interaction. The Ginzburg-Pitaevskii-Gross (GPG) nonlinear Schrodinger equation is extended to include an effective potential dependent on the square of the density and solved numerically for the s-wave. The lowest collective mode excitations are determined and their dependences on the number of atoms and on the strength of the three-body force are studied. The addition of three-body dynamics can allow the number of condensed atoms to increase considerably, even when the strength of the three-body force is very small compared with the strength of the two-body force. We study in detail the first-order liquid-gas phase transition for the condensed state, which can happen in a critical range of the effective three-body force parameter.

Dynamics of Bose-Einstein condensates in cigar-shaped traps

The Gross-Pitaevskii equation for a Bose-Einstein condensate confined in an elongated cigar-shaped trap is reduced to an effective system of nonlinear equations depending on only one space coordinate along the trap axis. The radial distribution of the condensate density and its radial velocity are approximated by Gaussian functions with real and imaginary exponents, respectively, with parameters depending on the axial coordinate and time. The effective one-dimensional system is applied to a description of the ground state of the condensate, to dark and bright solitons, to the sound and radial compression waves propagating in a dense condensate, and to weakly nonlinear waves in repulsive condensate. In the low-density limit our results reproduce the known formulas. In the high-density case our description of solitons goes beyond the standard approach based on the nonlinear Schrodinger equation. The dispersion relations for the sound and radial compression waves are obtained in a wide region of values of the condensate density. The Korteweg-de Vries equation for weakly nonlinear waves is derived and the existence of bright solitons on a constant background is predicted for a dense enough condensate with a repulsive interaction between the atoms.

Stabilization of a light bullet in a layered Kerr medium with sign-changing nonlinearity

Using the numerical solution of the nonlinear Schrodinger equation and a variational method, it is shown that (3+1)-dimensional spatiotemporal optical solitons, known as light bullets, can be stabilized in a layered Kerr medium with sign-changing nonlinearity along the propagation direction.

Dimensional reduction of a binary Bose-Einstein condensate in mixed dimensions

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Correlation times in stochastic equations with delayed feedback and multiplicative noise

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Permutability of Backlund transformations for N=2 supersymmetric sine-Gordon

The permutability of two Backlund transformations is employed to construct a nonlinear superposition formula and to generate a class of solutions for the N=2 super sine-Gordon model. We present explicitly the one and two soliton solutions.

The modulational instability in deep water under the action of wind and dissipation

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)