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In this paper, an exact unitary transformation is examined that allows forthe construction of solutions of coupled nonlinear Schr{\"o}dinger equationswith additional linear field coupling, from solutions of the problem where thislinear coupling is absent. The most general case where the transformation isapplicable is identified. We then focus on the most important special case,namely the well-known Manakov system, which is known to be relevant forapplications in Bose-Einstein condensates consisting of different hyperfinestates of $^{87}$Rb. In essence, the transformation constitutes a distributed,nonlinear as well as multi-component generalization of the Rabi oscillationsbetween two-level atomic systems. It is used here to derive a host of periodicand quasi-periodic solutions including temporally oscillating domain walls andspiral waves.Comment: 6 pages, 4 figures, Phys. Rev. A (in press

Linearly coupled Bose-Einstein condensates: From Rabi oscillations and quasiperiodic solutions to oscillating domain walls and spiral waves