  In this work we employ the split-step technique combined with a Legendre
pseudospectral representation to solve various time-dependent Gross-Pitaevskii
equations (GPE). Our findings based on the numerical accuracy of this approach
applied for one-dimensional (1D) and two-dimensional (2D) problems show that it
can provide accurate and stable solutions. Moreover, this approach has been
applied to study the dynamics of the Bose-Einstein condensate which is modeled
with the GPE. The breathing of condensate with an repulsive and attractive
interactions trapped in 1D and 2D harmonic potentials has been simulated as
well.
