Condensation of classical nonlinear waves in a two-component system

We study the formation of large-scale coherent structures (condensates) for a system of two weakly interacting Bose gases in the semiclassical approximation. Using the coupled defocusing nonlinear Schrödinger (NLS) equations as a representative model, we focus on condensation in the phase mixing regime. We employ weak turbulence theory to provide a complete thermodynamic description of the classical condensation process. We show that the temperature and the condensate mass fractions are fully determined by the total number of particles in each component and the initial total energy. Moreover, we find that, at higher energies, condensation can occur in only one component. We derive an analytic result for the variation of the critical energy where this transition occurs. The theory presented provides excellent agreement with results of numerical simulations obtained by directly integrating the dynamical model.