Numerical analysis of parametrized nonlinear equations
Numerical analysis of parametrized nonlinear equations
Numerical continuation methods: an introduction
Numerical continuation methods: an introduction
Numerical methods for bifurcations of dynamical equilibria
Numerical methods for bifurcations of dynamical equilibria
Introduction to Numerical Continuation Methods
Introduction to Numerical Continuation Methods
Liapunov-Schmidt Reduction and Continuation for Nonlinear Schro¨dinger Equations
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
Computing Ground States of Spin-1 Bose-Einstein Condensates by the Normalized Gradient Flow
SIAM Journal on Scientific Computing
Journal of Computational Physics
Journal of Computational and Applied Mathematics
Journal of Computational Physics
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We describe a novel two-parameter continuation method combined with a spectral-collocation method (SCM) for computing the ground state and excited-state solutions of spin-1 Bose-Einstein condensates (BEC), where the second kind Chebyshev polynomials are used as the basis functions for the trial function space. To compute the ground state solution of spin-1 BEC, we implement the single parameter continuation algorithm with the chemical potential @m as the continuation parameter, and trace the first solution branch of the Gross-Pitaevskii equations (GPEs). When the curve-tracing is close enough to the target point, where the normalization condition of the wave function is going to be satisfied, we add the magnetic potential @l as the second continuation parameter with the magnetization M as the additional constraint condition. Then we implement the two-parameter continuation algorithm until the target point is reached, and the ground state solution of the GPEs is obtained. The excited state solutions of the GPEs can be treated in a similar way. Some numerical experiments on Na23 and Rb87 are reported. The numerical results on the spin-1 BEC are the same as those reported in [10]. Further numerical experiments on excited-state solutions of spin-1 BEC suffice to show the robustness and efficiency of the proposed two-parameter continuation algorithm.