Mean-field theory of the atom-molecule Bose-Einstein condensate (AMBEC) is derived using the photoassociation approach to the atom-molecule coupling. The basis of the approach is the analogy between the coherent optical transient in the few-level system and the bound-free transition of the pair of atoms. Using the quasi-continuum set of the free states, the atom-molecule coupling is determined and used in the field theory of the photoassociation. The approximations that lead to the mean-field model of the simple AMBEC are presented and discussed. In the simple AMBEC it is shown that the s-wave atom-atom scattering length a becomes a tunable parameter. It is demonstrated that the mean-field amplitudes do not collapse, irrespective of sign of a. The collapse of a single condensate is in the simple AMBEC replaced by an instability.

Its origin lies in the peculiar property of the AMBEC Hamiltonian: the eigenstates are discontinuous along boundary lines termed fractures, and the collapse is a dynamical instability associated with crossing the fracture. The fractures in the two most important eigenstates, the ground state and the twin state, are calculated both numerically and analytically. The features of the instability and its location within the parameter space of the AMBEC, are shown to be in good qualitative agreement with the available experimental results.