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

Abstract

In this paper, we consider the classical Riemann problem for a generalized Burgers equation, u t + h α ( x ) u u x = u x x , $$\begin{equation*} u_t + h_{\alpha }(x) u u_x = u_{xx}, \end{equation*}$$ with a spatially dependent, nonlinear sound speed, h α ( x ) ≡ ( 1 + x 2 ) − 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 = 1 2 $\alpha =\frac{1}{2}$ .