Abstract
In this paper, we discuss the different splitting approaches to numerically solve the Gross–Pitaevskii equation (GPE). The models are motivated from spinor Bose–Einstein condensate (BEC). This system is formed of coupled mean-field equations, which are based on coupled Gross–Pitaevskii equations. We consider conservative finite-difference schemes and spectral methods for the spatial discretisation. Furthermore, we apply implicit or explicit time-integrators and combine these schemes with different splitting approaches. The numerical solutions are compared based on the conservation of the L 2 -norm with the analytical solutions. The advantages of the novel splitting methods for large time-domains are based on the asymptotic conservation of the solution of the soliton’s applications. Furthermore, we have the benefit of larger local time-steps and therefore obtain faster numerical schemes.
Highlights
The Bose–Einstein condensate (BEC) is an actual modelling problem for theoretical and experimental studies, see Reference [1]
The Gross–Pitaevskii equation is used to model the evolution of the Bose–Einstein condensate (BEC) order parameter for weakly interacting bosons, see References [2,3,4]
The mean field theory of BEC is based on coupled Gross–Pitaevskii equations with cubic nonlinear terms
Summary
The Bose–Einstein condensate (BEC) is an actual modelling problem for theoretical and experimental studies, see Reference [1]. The Gross–Pitaevskii equation is used to model the evolution of the Bose–Einstein condensate (BEC) order parameter for weakly interacting bosons, see References [2,3,4]. The mean field theory of BEC is based on coupled Gross–Pitaevskii equations (nonlinear Schrödinger equations) with cubic nonlinear terms These models allow us to predict matter-wave solitons in different configurations of condensates with attractive and repulsive interaction terms, see Reference [5]. A solitary wave or soliton solution is a localised travelling wave solution that retains its size, shape and speed when it moves. It does not spread or disperse, see Reference [8]. The numerical methods should have conservational behaviours to solve such a specialised balance, in the following two parts:
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