The article offers an analytical description of the nonlinear dynamics of the formation of regular impurity superstructures during the rapid crystallization of binary alloys, supplemented by numerical model calculations. The existence of a stable limit cycle is shown and the spatial profiles of the impurity concentration in a solid product are calculated. It has been established that considering the finiteness of the velocity of jumps of impurity atoms leads to a very significant decrease in the size of the cycles and a change in the profiles of the impurity superstructures.

The aim of the work is to present the conditions necessary for the appearance of layered periodic impurity superstructures as a result of solid-state phase transformations accompanied by a change in the impurity concentration in the general analytical form. Explicit formulas for the region of the values of the temperature gradient and the average velocity of the interphase boundary are obtained in the framework of the capillary-wave model. A formula is derived for the spatial period of the impurity superstructure, the appearance of which is due to these oscillations

The self-oscillation dynamic of an interface is calculated in the framework of the one-dimensional capillary-wave model. The parameters of a periodic layered impurity microstructure are obtained. This microstructure arises as a result of a solid-state phase transition with a change in concentration that occurs when a moving temperature gradient is applied.

The nonlinear differential equation described a dynamics of solid-phase explosive crystallization front in a much larger parameters domain in comparison with the theoretical results available in literature was obtained. The features of the self-oscillating mode transition of the front motion to the mode of its self-propagation with a constant velocity was numerically studied in detail.