A model of phase transitions with coupling between the order parameter and its gradient

A model of phase transitions with coupling between the order parameter and its gradient is proposed. It is shown, that this nonlinear model is suitable for the description of phase transitions accompanied by the formation of spatially inhomogeneous distributions of the order parameter. Exact solutions of the proposed model are obtained for the special cases which can be related to the spinodal decomposition or cosmological scenario. The proposed model is analogical to the mechanical nonlinear oscillator with the coordinate-dependent mass or velocity dependent elastic module. Based on this analogy, the existence of the limit cycles is established.