Moving planes for domain walls in a coupled system

The system leading to phase segregation in two-component Bose-Einstein condensates can be generalized hyperfine spin states with a Rabi term coupling. This leads domain wall solutions having monotone structure for non-cooperative system. We use the moving plane method prove monotonicity and one-dimensionality of transition solutions. relies on totally new estimates type which no Maximum Principle priori holds. also derive that one dimensional are unique up translations. When coefficient is large, we non-constant exist.