Exact dark state solutions of the coupled atomic-molecular Bose-Einstein condensates in an external potential

We consider a coupled nonlinear Schr\"{o}dinger equations describing an atomic Bose-Einstein condensates coupled to a molecular condensates through the stimulated Raman adiabatic passage loaded in an external potential. The existence of dark state are investigated within the full parameter space accounts for all the nonlinear collisions, together with the atom-molecule conversion coupling and external potential. The results show that there are a class of external potentials such as double-well, periodical and double periodical, and harmonic potentials, where the exact dark solutions can be formed. Our results may raise the possibility of relative experiments for dark state in the coupled atomic-molecular Bose-Einstein condensates.