The large-time development of the solution to an initial-value problem for the generalized Burgers equation

In this article, we consider an initial-value problem for the generalized Burgers’ equation. The normalized Burgers’ equation considered is given by ut+tδuux=uxx,−∞<x<∞,t>0, ut+tδuux=uxx,−∞<x<∞,t>0, where −12≤δ≠0−12≤δ≠0, and xx and tt represent dimensionless distance and time respectively. In particular, we consider the case when the initial data has a discontinuous step, where u(x,0)=u+u(x,0)=u+ for x≥0x≥0 and u(x,0)=u−u(x,0)=u− for x<0x<0, where u+u+ and u−u− are problem parameters with u+≠u−u+≠u−. The method of matched asymptotic coordinate expansions is used to obtain the large-tt asymptotic structure of the solution to this problem, which exhibits a range of large-tt attractors depending on the problem parameters. Specifically, the solution of the initial-value problem exhibits the formation of (i) an expansion wave when δ>−12δ>−12 and u+>u−u+>u−, (ii) a Taylor shock (hyperbolic tangent) profile when δ>−12δ>−12 and u+<u−u+<u− and (iii) the Rudenko–Soluyan similarity solution when δ=−12δ=−12.