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

Abstract

AbstractIn this paper, we consider the classical Riemann problem for a generalized Burgers equation, with a spatially dependent, nonlinear sound speed, with , which decays algebraically with increasing distance from a fixed spatial origin. When , 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 , 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 .