Preface Chapter 1 Introduction 1.1 Brief history of Bose-Einstein condensation 1.2 Quantized vortex states in BEC 1.3 Review on numerical methods for stationary states 1.4 Review on numerical methods for the time-dependent GPE 1.5 Scope of this book Chapter 2 Stationary states for rotating BEC 2.1 GPE in a rotational frame 2.1.1 Dimensionless GPE 2.1.2 Reduction to two dimensions 2.2 Stationary states 2.2.1 Semiclassical scaling and geometrical optics 2.2.2 Ground state 2.2.3 Approximate ground state 2.2.4 Excited states 2.2.5 Critical angular velocity in symmetric trap 2.3 Numerical methods for stationary states 2.3.1 Gradient flow with discrete normalization 2.3.2 Energy diminishing 2.3.3 Continuous normalized gradient fiow 23.4 Fully numerical discretization 2.4 Numerical results 2.4.1 Initial data for computing ground state 2.4.2 Results in 2D 2.4.3 Results in 3D 2.4.4 Critical angular velocity 2.4.5 Numerical verification for dimension reduction 2.4.6 Errors of the TF approximation 2.4.7 Spurious numerical ground states when 2.5 Conclusion Chapter 3 Dynamics of rotating BEC 3.1 Some properties of the GPE 3.2 A TSSP method for the GPE 3.2.1 Time-splitting 3.2.2 Discretization in 2D 3.2.3 Discretization in 3D 3.2.4 Stability 3.3 Numerical results 3.3.1 Accuracy test 3.3.2 Dynamics of a vortex lattice in rotating BEC 3.3.3 Generation of giant vortex in rotating BEC 3.4 Conclusion N Chapter 4 Applications to stationary states of rotating two-component BEC 4.1 The time-dependent coupled GPEs 4.1.1 Dimensionless coupled GPEs 4.1.2 Reduction to two dimensions 4.1.3 Semiclassical scaling 4.2 Stationary states 4.2.1 Ground state 4.2.2 Symmetric and central vortex states 4.2.3 Numerical methods for the stationary states 4.2.4 Numerical results for the stationary states 4.3 Conclusion Chapter 5 Applications to dynamics of rotating two-component BEC 5.1 Some properties of the coupled GPEs 5.2 A TSSP method for the coupled GPEs 5.2.1 Time-splitting 5.2.2 Discretization in 2D 5.2.3 Stability 5.3 Numerical results for the dynamics 5.4 Conclusion Chapter 6 Application into ground state of spinor BEC 6.1 A continuous normalized gradient flow 6.1.1 Euler-Lagrange equations 6.1.2 A continuous normalized gradient flow 6.2 Normalization and magnetization conservative and energy diminishing numerical scheme 6.2.1 Semi-discretization in time …… Chapter 7 Applications to dynamics of spinor F=1BEC Chapter 8 Concluding remarks and future work Bibliography
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