Programa de Pós-graduação em Física

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Estudo de soluções localizadas na equação não linear de Schrödinger logarítmica, saturada e com efeitos de altas ordens
(Universidade Federal de Goiás, 2018-06-07) Alves, Luciano Calaça; Avelar, Ardiley Torres; http://lattes.cnpq.br/5732286631137637; Cardoso, Wesley Bueno; http://lattes.cnpq.br/6845416823133684; Bazeia Filho, Dionisio; Valverde, Clodoaldo; Santana, Ricardo Costa de; Maia, Lauro June Queiroz
This work presents the study of solitary wave solutions, known as solitons, in non-linear and non- homogeneous media using non-linear Schrödinger equations. Three cases are studied: first considering a logarithmic nonlinear term; second with saturation effect and finally including effects of high orders (Raman scattering). Solutions are modulated by three different types of potential. First, linear in the spatial and oscillatory coordinate in the temporal coordinate. The second, quadratic in the spatial and oscillatory in the temporal coordinates. Finally, it is also modulated using a mixed potential, which is the junction of the two potentials presented above. After including inomogeneities in linear and nonlinear coefficients, the similarity transformation technique is used to convert the non-linear, non-autonomous equation into an autonomous one that will be solved analytically. This field of study has potential applications in crystals, optical fibers and in Bose- Einstein condensates, also serving to understand the fundamentals related to this state of matter. The stability of the solutions are checked by numerical simulations.
Dinâmica e decaimento de sólitons escuros em condensados de Bose-Einstein atômicos quase-unidimensionais
(Universidade Federal de Goiás, 2014-02-25) Couto, Hugo Leonardo Carvalhaes; Cardoso, Wesley Bueno; Cardoso, Wesley Bueno; Losano, Laercio; Avelar, Ardiley Torres
(Sem resumo em outra língua)
Método compacto de diferenças finitas para resolver equações de Schrödinger não lineares com dispersão de quarta ordem
(Universidade Federal de Goiás, 2016-09-16) Jesus, Hugo Naves; Cardoso, Wesley Bueno; http://lattes.cnpq.br/6845416823133684; Cardoso, Wesley Bueno; http://lattes.cnpq.br/6845416823133684; Avelar, Ardiley Torres; Bazeia, Dionisio; Mendanha Neto, Sebastião Antônio
Finite difference schemes belong to a class of numerical methods used to approximate derivatives. They are widely used to find approximations to differential equations. There are a lot of numerical methods, whose deductions are made through expansions in Taylor Series. Depending on the manner in which expansion is made, it can be combined with other expansions to obtain derivatives with better numerical approximations. Usually when we get numerical derivative with better approaches, it is necessary to increase the amount of points used in the grid. An alternative to this problem are compact methods, which achieve better approximations for the same derivative but without increasing the number of mesh points. This work is an attempt to develop the Compact-SSFD method for the Schrödinger Equation Nonlinear Fourth Order. SSFD methods are used to separate the parts of a differential equation so that each part can be solved separately. For example in the case of non-linear differential equations it is often used to separate the linear parts of nonlinear parts. In Compact-SSFD methods nonlinear parts are resolved exactly as the linear are resolved using compact methods. Our work is inspired in the Dehghan and Taleei’s work where was used the Compact-SSFD method for solving numerically the equation Nonlinear Schrödinger. Before we try to develop our method, the results of the authors was correctly reproduced. But when we try to deduce a method analogous to the differential equation we wanted to solve, which also involves derived from fourth order, we realized that a Compact type method does not get as trivially as in the case of used to approach second-order derivatives.
Quebra de simetria em condensados de Bose-Einstein confinados por um potencial funil
(Universidade Federal de Goiás, 2021-04-09) Miranda, Bruno Martins; Cardoso, Wesley Bueno; http://lattes.cnpq.br/6845416823133684; Cardoso, Wesley Bueno; Malbouisson, Jorge Mário Carvalho; Almeida, Norton Gomes de
Theoretically predicted in 1923-1924 by Bose and Einstein and experimentally obtained only in 1995, the Bose-Einstein condensate became an important laboratory for the investigation of various quantum phenomena, such as the Josepshon oscillations, the study of vortex, use as interferometers, etc. Using mean-field theory to include the effects of the average interaction between particles, in the 1960s, Gross and Pitaevskii obtained an equation capable of describing the dynamics of a diluted gas at a temperature of 0 K. Dimensional reduction models for the Gross-Pitaevskii equation were developed for several types of confining potentials in order to simplify numerical calculations and reproduce accurate results. For condensates with a strong attractive strength, confined by doublewell potentials, it is known that the phenomenon of spontaneous symmetry breaking occurs. In this state, the particle population between wells becomes asymmetrical, in contrast with the symmetry of the confining potential. In this work, we consider a condensate in the self-focusing regime, confined transversely by a funnel-like potential and axially by a double well formed by the combination of two inverted Pöschl-Teller potentials. We used an effective equation, obtained by means of a variational method for the Gross-Pitaevskii equation, to analyze the symmetry break of the probability density of the wave function that describes the condensate. This symmetry break was observed for several interaction strength values as a function of the minimum potential well. A quantum phase diagram was obtained, in which it is possible to recognize the three phases of the system: symmetric phase (Josepshon), asymmetric phase (spontaneous symmetry breaking - SSB), and collapsed states, i.e., when the solution becomes singular, which does not represent the physical system, showing a validity limit for the model under consideration. We analyzed our symmetric and asymmetric solutions using the real-time evolution method, in which it was possible to confirm the stability of the results. Finally, a comparison with the cubic nonlinear Schrödinger equation in one dimension and the Gross-Pitaevskii equation in three dimensions is performed for the purpose of analyzing the accuracy of the effective equation used here.
Redução dimensional para condensados de Bose-Einstein em forma de “tubo” e “anilha plana”
(Universidade Federal de Goiás, 2019-02-28) Santos, Mateus Calixto Pereira dos; Cardoso, Wesley Bueno; http://lattes.cnpq.br/6845416823133684; Cardoso, Wesley Bueno; Avelar, Ardiley Torres; Santana, Ademir Eugênio de
The study of nonlinear dynamics represents a challenge of contemporary physics. In particular, the investigation of Bose Einstein condensates proved to be a hard task due to the large number of interacting particles. Therefore, given the difficulty of modeling these systems, approximations were introduced, which promoted the description of the Bose-Einstein condensation state in interacting atomic gases as a three-dimensional nonlinear Schrödinger equation, known as the Gross-Pitaevskii equation. In this work we review the dimensional reduction method, which use a variational treatment with the goal of derive effective one-dimensional (1D) and two-dimensional (2D) equations in cigar-shaped and pancake-shaped Bose-Einstein condensates, where we show that these equations describe almost exactly the dynamics of their respective models. Thus, we studied the ground-state solutions in tube-shaped and flat washer-shaped Bose-Einstein condensates by means of effectives non-polynomials equations, derived from the dimensional reduction method. The results produced by this equations were in very good agreement with those obtained from the corresponding full 3D Gross-Pitaevskii equation.
Espalhamento de gaussons em acopladores ópticos direcionais
(Universidade Federal de Goiás, 2017-04-10) Teixeira, Rafael Marques Paes; Cardoso, Wesley Bueno; http://lattes.cnpq.br/6845416823133684; Avelar, Ardiley Torres; Ramos, Jorge Gabriel Gomes de Souza
This dissertation presents the research that was focused on the study of solitons in nonintegrable systems governed by effective models of partial differential equations (PDEs) of (1+1) dimensions, that is, one spatial and one time dimension. In the main research line, we considered a model of two fields given by nonlinear Schrödinger equations with logarithmic nonlinearity and linear coupling, the main aspects of the collision dynamics of two Gaussian solitons (gaussons) were investigated, addressing the stability of the solutions, the interaction mechanisms, and the regularity of the scattering. The study of gaussons collisions in this model was performed via two approaches: semianalytic (variational/reduced model) and direct numerical simulation. For the system considered in this work, both approaches provided similar results which revealed the existence of a chaotic and fractal dynamics involving the initial conditions and the post-collisional properties of the solitons, which were verified by the analysis of the correlations between the input and output parameters. The fractal patterns are constituted by regularity windows that are positioned accordingly to a well defined rule, which was obtained via a linearization model that was found to be very similar in both approaches (i.e., the windows arrangement was well described by the variational model). In such windows the solitons collision dynamic turns out to be not chaotic (regular collision), it becomes predictable and very well defined, hence the width of these windows provides the conditions in which the control of the interaction is possible. Besides, by analyzing the dynamics of the variational parameters of the reduced model and also the dynamics of the most relevant physical quantities in the exact approach via PDEs (namely the total kinetic and potential energies), we attested that these windows arise due to a resonant mechanism of energy exchange between translational and vibrational soliton modes. By studying the linear stability of the solitary waves, we proved that the vibrational modes originate from shape oscillations induced by internal modes and also, possibly, by instability modes excited during the collisions. We verified that these oscillations are well defined in all regular collisions. The viability of the variational method in the system studied is due to the suitable description of the vibrational modes by the reduced model, and also the low influence of the emitted nonlinear radiation in the regular collision processes found in the direct simulations.

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