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Sentation respectively. Therefore, one may employ the so-called split-step technique [3] to
decompose the original problem into two subproblems and solve them in real-space and
momentum-space respectively. The main idea of split-step is following: Let’s write a gen-
eral evolution equation in the form

iω_t =    + .√L)ω

where J? and ʃ are momentum-dependent and space-dependent operators, respectively.
For the GP equation shown above we have

⅛2π2 pl

= -ʒ- = ɔ ʃ = [Vext(r) + g∣ψ(r, Ol²]               (2.3)

2m Im

The basic procedure of the split-step method is to approximate the solution of Eq.(2.2) by
solving the purely space-dependent and purely momentum-dependent equation in a given
sequential order, and the solution of the previous subproblem is employed as an initial
condition for the consequent subproblem. We represent here the first-order splitting scheme
for a GP equation as an example.

Advancing     ∙, _ιz._ι,        FT ~             Advancing ∙_o?.. ~            inverseFl'

Ψ,∙-----→ e’^t) -→ 4>_i(t + ∆t)-----→         + ∆t)------> Ψ,∙(t + ∆t) (2.4)

As we can see above, in split-step scheme, we only need to work with dignonal matrices
in each step by swapping back and forth from real space to momentum space, which makes
this scheme highly efficient. The result given by split-step scheme is very accurate in space
and has the second-order accuracy in time. The splitting error in time stems from the

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