*I-Liang Chern*

*Professor Department of Mathematics National Taiwan University*

In this talk, I will first give a brief introduction to the spinor Bose-Einstein condensates (BECs). Then I will present two recent results, one is numerical, the other is analytical for spinor BECs w/o uniform external magnetic field. In the numerical study of spinor BECs, a pseudo-arclength continuation method (PACM) was proposed for investigating the ground state patterns and phase diagrams of the spin-1 Bose-Einstein condensates under the influence of a homogeneous magnetic field. Two types of phase transitions are found. The first type is a transition from a two-component (2C) state to a three-component (3C) state. The second type is a symmetry breaking in 3C state. After that, a phase separation of the spin component occurs. In the semi-classical regime, these two phase transition curves are gradually merged.

In the analytical study, the ground states of spin-1 BEC are characterized. First, we present the case when there is no external magnetic field. For ferromagnetic systems, we show the validity of the so-called single-mode approximation (SMA). For antiferromagnetic systems, there are two subcases. When the total magnetization Mâ‰ 0, the corresponding ground states have vanishing zeroth (*mF* = 0) components (so call 2C state), thus are reduced to two-component systems. When M=0, the ground states are also reduced to the SMA, and there are one-parameter families of such ground states. Next, we study the case when an external magnetic field is applied. It is shown analytically that, for antiferromagnetic systems, there is a phase transition from 2C state to 3C state as the external magnetic field increases. The key idea in the proof is a redistribution of masses among different components, which reduces kinetic energy in all situations, and makes our proofs simple and unified. The numerical part is a joint work with Jen-Hao Chen and Weichung Wang, whereas the analytical part is jointly with Liren Lin.