One-dimensional cubic-quintic Gross-Pitaevskii equation for Bose-Einstein condensates in a trap potential

Abstract

By means of new general variational method we report a direct solution for the quintic self-focusing nonlinearity and cubic-quintic 1D Gross Pitaeskii equation (GPE) in a harmonic confined potential. We explore the influence of the 3D transversal motion generating a quintic nonlinear term on the ideal 1D pure cigar-like shape model for the attractive and repulsive atom-atom interaction in Bose Einstein condensates (BEC). Also, we offer a closed analytical expression for the evaluation of the error produced when solely the cubic nonlinear GPE is considered for the description of 1D BEC.

Validity of Gross–Pitaevskii solutions of harmonically confined BEC gases in reduced dimensions

“…On the other hand, as we shall argue below, this physical picture is not quite simply obeyed by the Gross-Piatevskii equation. In any, case, we follow here the procedure delineated in [32,34], in which by properly choosing the sizes of these frequencies, the dynamics along different dimensions can be decoupled. This decoupling allows for an adiabatic approximation such that in the tight direction, namely where ω x =ω y ?…”

“…This implies that the non-linear dynamics occurs solely in the directions where the frequencies are small. As briefly discussed below, we consider two ways in which the adiabatic component can be integrated out, one at the level of the energy functional [32], and the other at the level of the GP equation itself [34]. This leads to a small but noticeable difference in the effective nonlinear coupling constant in the respectively obtained 1D and 2D dynamical GP equations.…”

“…This procedure, however, requires a theoretical justification that both yields the correct equation in the reduced dimension and sets their bounds of validity. This theoretical procedure, of how to conduct such a dimension reduction, has been the subject of several studies [28][29][30][31][32][33][34][35][36][37][38], with the most important result that the atomic interaction strength becomes modified in 1D and 2D with respect to the original interaction in 3D, usually being scaled by the length of the oscillator in the constrained direction or directions. In some of those cases [29][30][31], however, the reduced equations show a non-linearity different from the usual one in GP equations.…”

“…We proceed by, first, briefly describing the two main different approaches [32,34] that deal with the reduction of dimensionality of the 3D GP equation to 2D and 1D effective equations. While differing in the precise step where the reduction is enabled, both procedures are based on a variational scheme followed by an adiabatic approximation which assumes that the dynamics is spatially decoupled; in this way, the component of the gas in the squeezed or small dimension or dimensions stays in its non-interacting ground state, and the atomic collisions, represented by non-linear terms, take place in the desired reduced dimension.…”

Exact soliton solutions of the generalized Gross-Pitaevskii equation based on expansion method

“…44,45 To eliminate the interdependence between k(t) and g(t) and allow them vary freely, we introduce a parameter function σ (t) into the coefficients of the above equation through the following transformation…”