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首页> 外文期刊>Studies in Applied Mathematics >The Riemann problem for a generalized Burgers equation with spatially decaying sound speed. I Large‐time asymptotics
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The Riemann problem for a generalized Burgers equation with spatially decaying sound speed. I Large‐time asymptotics

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

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Abstract 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→∞$trightarrow infty$, 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$alpha =frac{1}{2}$.

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