Application of Particular Solutions of the Burgers Equation to Describe the Evolution of Shock Waves of Density of Elementary Steps

Abstract

Particular solutions of the Burgers equations (BE) with zero boundary conditions are investigated in an analytical form. For values of the shape parameter  greater than 1, but approximately equal to 1, the amplitude of the initial periodic perturbations depends nonmonotonically on the spatial coordinate, i.e. the initial perturbation can be considered as a shock wave. Particular BE solutions with zero boundary conditions describe a time decrease of the amplitude of initial nonmonotonic perturbations, which indicates the decay of the initial shock wave. At large values of the shape parameter , the amplitude of the initial periodic perturbations depends harmoniously on the spatial coordinate. It is shown that over time, the amplitude and the spatial derivative of the profile of such a perturbation decrease and tend to zero. Emphasis was put on the fact that particular BE solutions can be used to control numerical calculations related to the BE-based description of shock waves in the region of large spatial gradients, that is, under conditions of a manifold increase in spatial derivatives. These solutions are employed to describe the profile of a one-dimensional train of elementary steps with an orientation near <100>, formed during the growth of a NaCl single crystal from the vapor phase at the base of a macroscopic cleavage step. It is shown that the distribution of the step concentration with distance from the initial position of the macrostep adequately reflects the shock wave profile at the decay stage. The dimensionless parameters of the wave are determined, on the basis of which the estimates of the characteristic time of the shock wave decay are made.