A Two-Parameter Continuation Method for Rotating Two-Component Bose-Einstein Condensates in Optical Lattices

Abstract

We study efficient spectral-collocation and continuation methods (SCCM) for rotating two-component Bose-Einstein condensates (BECs) and rotating two-component BECs in optical lattices, where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. A novel two-parameter continuation algorithm is proposed for computing the ground state and first excited state solutions of the governing Gross-Pitaevskii equations (GPEs), where the classical tangent vector is split into two constraint conditions for the bordered linear systems. Numerical results on rotating two-component BECs and rotating two-component BECs in optical lattices are reported. The results on the former are consistent with the published numerical results.

Efficient spectral collocation and continuation methods for symmetry-breaking solutions of rotating Bose–Einstein condensates

“…For instance, the two components in rotating two-component BECs have opposite vortex structures. See [33] and the further references cited therein. The numerical results in Example 4 show that the two components also have different nodal line structures for symmetry-breaking solutions.…”

“…We used a two-parameter continuation algorithm [33] for computing symmetry-breaking solutions of Equation (56), where the chemical potentialsλ 1 andλ 2 are treated as the continuation parameters simultaneously. Here, we chose η 11 = 800, η 12 = η 21 = 400, and η 22 = 500.…”

Multi-parameter continuation and collocation methods for rotating multi-component Bose–Einstein condensates

“…Wang et al [29] proposed a two-parameter continuation algorithm for rotating two-component BECs, where the chemical potentials λ 1 and λ 2 associated with the two components ψ 1 and ψ 2 were treated as the continuation parameters simultaneously. Accordingly the original tangent vectors were split into two unit tangent vectors which were used as the constraint conditions for solving the bordered linear systems.…”

“…First, we consider the two-component case. we use the two-parameter predictor-corrector continuation algorithm [29] to trace the ground state solutions of Equation (3) with ω = 0, where the chemical potentials λ 1 and λ 2 associated with the wave functions ψ 1 and ψ 2 are treated as the continuation parameters simultaneously, and the original tangent vector is split into two unit tangent vectors which are used as the constraint conditions for the bordered linear systems. As one of the two components, say, ψ 1 satisfies the normalization condition, we finish the first stage curve-tracking, and obtain the first target point.…”

Acceleration of the imaginary time method for spectrally computing the stationary states of Gross–Pitaevskii equations

“…It is admitted that deriving robust and efficient numerical approaches for the stationary state computation is difficult, most particularly when the nonlinearity is large and the rotation velocity is high. More generally, methods are based either on solving the nonlinear eigenvalue problem [27,37,38] or deriving nonlinear optimization techniques under constraints [14,18,19,24,25]. A standard alternative approach is the Imaginary Time Method (ITM) also called Continuous Normalized Gradient Flow (CNGF) [2,6,8,11,13,15,21,22,41].…”