Abstract
We study the existence of nontrivial solution branches of three-coupled Gross–Pitaevskii equations (CGPEs), which are used as the mathematical model for rotating spin-1 Bose–Einstein condensates (BEC). The Lyapunov–Schmidt reduction is exploited to test the branching of nontrivial solution curves from the trivial one in some neighborhoods of bifurcation points. A multilevel continuation method is proposed for computing the ground state solution of rotating spin-1 BEC. By properly choosing the constraint conditions associated with the components of the parameter variable, the proposed algorithm can effectively compute the ground states of spin-1 ^{87}Rb and ^{23}Na under rapid rotation. Extensive numerical results demonstrate the efficiency of the proposed algorithm. In particular, the affect of the magnetization on the CGPEs is investigated.
Highlights
Experimental results on rotating spinor Bose–Einstein condensates (BEC)[1,2,3] have intrigued researchers on theoretical physics and applied mathematics to study quantum phenomena of superfluidity, such as hexagonal vortex lattice and square vortex lattice, which do not exist in a single component BEC
Based on the mean-field theory, the dynamics of rotating spinor BEC at ultra-cold temperatures can be described by the nonlinear Schrödinger equations (NLS), or the coupled Gross–Pitaevskii equations[12,13,14] (CGPEs) as follows: i ∂t φ1(x, t) =
We investigate the existence of nontrivial solution curves of the coupled Gross–Pitaevskii equations12–14 (CGPEs) using the Lyapunov–Schmidt reduction
Summary
Various numerical methods have been proposed for computing the ground state solution of both one- and two-component rotating BEC23–33. The ground state solution of rotating spin-1 BEC is obtained by minimizing the energy functional E( ) subjected to the constraints Eqs. Note that the numerical computations of the ground states for spin-1 BEC with quadratic Zeeman energy has been widely investigated in40. We will apply the proposed algorithm to study how the wave function of Eq (10) changes with respect to the angular velocity when ω > 1 , where we impose a harmonic plus quartic trap on the system. In section “Existence of nontrivial solution curves” we present the existence of nontrivial solution curves branching from bifurcation points of the CGPEs. A multi-parameter continuation algorithm is proposed in section “A multilevel continuation algorithm” for computing the ground state of (rapidly) rotating spin-1 BEC.
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