A basic two-state model for bosonic field theories with a cubic nonlinearity

A basic nonlinear two-state model generic in classical and bosonic field theories with a cubic nonlinearity is considered. For the class of models with constant external field amplitude a general strategy for attacking the problem is developed based on the reduction of the initial system of equations for the semi-classical atom-molecule amplitudes to a nonlinear Volterra integral equation for the molecular probability. A uniformly convergent series solution to the problem is constructed for the weak interaction limit. The Landau-Zener model is considered as a specific example. The first approximation term is derived and an asymptotic expression for the nonlinear transition probability is established in the weak interaction regime