  An approach is proposed to solve the coupled Gross-Pitaevskii equations (CGP)
of the 3-species BEC in an analytical way under the Thomas-Fermi approximation
(TFA). It was found that, when the strength of a kind of interaction increases
and crosses over a critical value, a specific type of state-transition will
occur and will cause a jump in the total energy. Due to the jump, the energy of
the lowest symmetric state becomes considerably higher. This leaves a
particular opportunity for the lowest asymmetric state to replace the symmetric
states as the ground state. It was further found that the critical values are
related to the singularity of either the matrix or a sub-matrix of the CGP.
These critical values are not arising from the TFA but inherent in the CGP, and
they can be analytically expressed. Furthermore, a model (in which two kinds of
atoms separated from each other asymmetrically) has been proposed for the
evaluation of the energy of the lowest asymmetric state. With this model the
emergence of the asymmetric ground state is numerically confirmed under the
TFA. The theoretical formalism of this paper is quite general and can be
generalized for BEC with more than three species.
