On the asymptotics of a solitary internal wave in the rarefaction wave mode

Transport flows, diffusion processes of new technologies, processes of filtration and gas dynamics are described by the first order quasilinear equation. The asymptotics of the solution of the Cauchy problem for this equation allows us to introduce characteristics of the dynamic process which are important from a substantive view point, for example, the diffusion rate of new technologies in diffusion models of Schumpeter type technologies. The convergence rate of the Cauchy problem solution to the asymptotics determines the characteristic times for these concepts to be used and depends on the internal waves dynamics. Using the method of characteristics, the paper studies the rate for the solution to reach the asymptotics for the first order quasilinear equation of the scalar conservation law.