The Riemann problem for a generalized Burgers equation with spatially decaying sound speed. I Large‐time asymptotics

In this paper, we consider the classical Riemann problem for a generalized Burgers equation, ut+hα(x)uux=uxx,$$\begin{equation*} u_t + h_{\alpha }(x) u u_x = u_{xx}, \end{equation*}$$with a spatially dependent, nonlinear sound speed, hα(x)≡(1+x2)−α$h_{\alpha }(x) \equiv (1+x^2)^{-\alpha }$ with α>0$\alpha >0$, which decays algebraically with increasing distance from a fixed spatial origin. When α=0$\alpha =0$, this reduces to the classical Burgers equation. In this first part of a pair of papers, we focus attention on the large‐time structure of the associated Riemann problem, and obtain its detailed structure, as t→∞$t\rightarrow \infty$, via the method of matched asymptotic coordinate expansions (this uses the classical method of matched asymptotic expansions, with the asymptotic parameters being the independent coordinates in the evolution problem; this approach is developed in detail in the monograph of Leach and Needham, as referenced in the text), over all parameter ranges. We identify a significant bifurcation in structure at α=12$\alpha =\frac{1}{2}$.