Abstract: In this paper, a normalized gradient flow with Lagrange multipliers is designed to compute ground states of spin-orbit coupled Spin-2 Bose-Einstein condensates. By excavating the implicit relation between projection coefficients, the difficult that the existed conditions (the conservation of total mass and magnetization) of the model problem is insufficient to determine all the projection coefficients, is overcome. Extensive numerical experiments are done to compute the ground states of spin-orbit coupled Spin-2 BECs with cyclic/ferromagenetic interactions. As a result, the effectiveness of the two algorithms is verified, and the phase transformation law about how the stripe pattern ground states and the square-lattice pattern ground states of spin-orbit coupled Spin-2 BECs change to each other with the spin-orbit coupling parameter, are revealed.

Key words: Bose-Einstein condensate, Ground state, Normalized gradient flow, Lagrange multiplier