| Summary: | 博士 === 國立清華大學 === 數學系 === 92 === In Chapter 1, we propose fixed point methods for computing the
energy state solutions of the time-independent vector Gross-Pitaevskii equation (VGPE) which describes a multi-component Bose-Einstein condensate. We prove that the fixed point iterative methods converge locally and linearly to a solution of the VGPE if and only if the associated minimized energy functional problem has a strictly local minimum. The iterative methods can also be used to compute the bifurcation diagram of ground states and bound states, as well as the energy functional. Numerical experience shows that our iterative methods converge globally and linearly in 10 to 20 steps. In particular, we observe a new phenomenon: verticillate multipling, i.e., the generation of multiple verticillate structures.
In Chapter 2, we derive the asymptotic motion equations of vortices for the time-dependent Gross-Pitaevskii equation with a
harmonic trap potential. The asymptotic motion equations form a
system of ordinary differential equations which can be regarded as a perturbation of the standard Kirchhoff problem. From the
numerical simulation on the asymptotic motion equations, we
observe that the bounded and collisionless trajectories of three
vortices form chaotic, quasi 2- or quasi 3-periodic orbits.
Furthermore, a new phenomenon of $1:1$-topological ynchronization is observed in the chaotic trajectories of two vortices.
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