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The article solves the problem of theoretically determining the deformable characteristics of railway ballast, considering its condition through mathematical modeling. Different tasks require mathematical models with different levels of detail of certain elements. After a certain limit, excessive detailing only worsens the quality of the model. Therefore, for many problems of the interaction between the track and the rolling stock, it is sufficient to describe the ballast as a homogeneous isotropic layer with a vertical elastic deformation. The elastic deformation of the ballast is formed by the deviation of individual elements; the ballast may have pollutants, the ballast may have places with different levels of compaction, etc. To be able to determine the general characteristics of the layer, a dynamic model of the stress–strain state of the system based on the dynamic problem of the theory of elasticity is applied. The reaction of the ballast to the dynamic load is modeled through the passage of elastic deformation waves. The given results can be applied in the models of the railway track in the other direction as initial data regarding the ballast layer.

In many countries around the world, a cubic parabola is used as transition curves in a simplified form. Today, the properties of many mathematical curves are well studied, which can be used as transition curves. The object of research is the process of restructuring transition curves when organizing high-speed trains operation. The use of modern means of mathematical modelling allows to add a number of dynamic characteristics for evaluation parameters of motion in the curved sections of the tracks. Thus, with an increase in speed for transition curves, both the values of unbalanced acceleration and its change in time, and also the velocity attenuation of oscillations of rolling stock when moving from a transition curve to a circular curve or to a direct section. On the basis of the conducted research, it has been found that when increasing the speed of the vehicle movement in the range from 120 to 200 km/h, the length of the stabilization section increases at the output of the curvature by an average of 20 m for every 40 km/h. The analysis showed that with correctly set parameters of transition curves, which correspond to the maximum speed of movement and radius of a circular curve, the strength and reliability of the bearing elements in the structure of the permanent way and the travel comfort are fully ensured.