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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 increasesand crosses over a critical value, a specific type of state-transition willoccur and will cause a jump in the total energy. Due to the jump, the energy ofthe lowest symmetric state becomes considerably higher. This leaves aparticular opportunity for the lowest asymmetric state to replace the symmetricstates as the ground state. It was further found that the critical values arerelated 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, andthey can be analytically expressed. Furthermore, a model (in which two kinds ofatoms separated from each other asymmetrically) has been proposed for theevaluation of the energy of the lowest asymmetric state. With this model theemergence of the asymmetric ground state is numerically confirmed under theTFA. The theoretical formalism of this paper is quite general and can begeneralized for BEC with more than three species.

Singularity in the matrix of the coupled Gross-Pitaevskii equations and the related state-transitions in three-species condensates