In this paper we present mathematical analysis of one-dimensional effective models proposed in [\cite{MunozDelgado}] concerning Bose-Einstein condensates in the presence of harmonic confinement. Among the demonstrated properties, we can mention: existence, uniqueness, orbital stability, symmetry and gaussian asymptotic decay of ground-state solutions in the repulsive case. We also report formulæ for the minimal energy $E_{\mn}$ and the associate chemical potential
μ as functions of a parameter
λ, which is related to
N (the number of atoms) and/or
a (the s-wave scattering length). By considering Taylor's development of the non-quadratic therm of the energy and using appropriate gaussian functions as approximations for the ground state, we present some numerical experiments to illustrate our results.
We study exact matter-wave solutions of the quasi-one-dimensional Gross-Pitaevskii (GP) equation with the space- and/or time-modulated potential and nonlinearity and the time-dependent gain or loss term in Bose-Einstein condensates. In particular, based on the similarity transformation and symbolic analysis, we report several families of exact solutions of the quasi-one-dimensional GP equation in the combination of the harmonic and Gaussian potentials, in which some physically relevant solutions are described. The stability of the obtained matter-wave solutions is addressed numerically such that some stable solutions are found. Moreover, we also analyze the parameter regimes for the stable solutions. These results may raise the possibility of relative experiments and potential applications.
In this paper we present a mathematical analysis of one-dimensional effective model proposed by A.M.~Mateo and V.~Delgado concerning Bose-Einstein condensates in the presence of harmonic confinement. Among the properties presented, we mention: existence, ``uniqueness'', orbital stability, symmetry and gaussian asymptotic decay of ground-state solutions in the repulsive case. We also report formul\ae\ for the minimal energy $E_{\mn}$ and the associate chemical potential
μ as functions of
λ, which is a parameter related to
N (the number of atoms) and/or
a (the s-wave scattering length). By considering Taylor's development of the non-quadratic therm of the energy and using appropriate gaussian functions as approximations for the ground state, we present some numerical experiments to illustrate our results