Solución Numérica de los Estados Base y Primer Excitado de Condensados de Bose-Einstein 1D y 2D Usando los Métodos de BESP Y BFSP.

Within this report we are describing the treatment made to the Gross-Pittaevsky equation, which describes the behavior of Bose-Einstien condensates, in order to be normalized and nondimensionalized. We show that the condensate solutions are those normalized functions for which the energy functional minimices. In order to solve the problem the equation is discretized by means of a gradient flow (GFDN) which, under certain circumstances, is energy diminishing and therefore appropriate to find ground and first excited states under proper initial conditions. To discretize completely the GFDN two algorithms known as Backward Euler sine-pseudospectral method (BESP) and Backward/Forward Euler sine-pseudospectral method (BFSP) are proposed. In the framework of this methods the spacial derivatives of the wave function can be written in terms of sine transforms which make easier the solution of the linear system appearing when the iteration is carried out. Finally we report the results obtained by the simulations of condensates and we find some properties of the dependence of the wave function and its response to different potentials with respect to the condensate interaction parameter. [ABSTRACT FROM AUTHOR]