The large-time development of the solution to an initial-value problem for the generalised burgers’ equation

Abstract

In this article, we consider an initial-value problem for the generalized Burgers’ equation. The normalized Burgers’ equation considered is given bywhere $-\frac{1}{2} \le \delta \not= 0$ , and $x$ and $t$ represent dimensionless distance and time respectively. In particular, we consider the case when the initial data has a discontinuous step, where $u(x,0)= u_{+}$ for $x \ge 0$ and $u(x,0)=u_{-}$ for $x<0$ , where $u_{+}$ and $u_{-}$ are problem parameters with $u_{+} \not= u_{-}$ . The method of matched asymptotic coordinate expansions is used to obtain the large- $t$ asymptotic structure of the solution to this problem, which exhibits a range of large- $t$ attractors depending on the problem parameters. Specifically, the solution of the initial-value problem exhibits the formation of (i) an expansion wave when $\delta>-\frac{1}{2}$ and $u_{+}>u_{-}$ , (ii) a Taylor shock (hyperbolic tangent) profile when $\delta>-\frac{1}{2}$ and $u_{+}<u_{-}$ and (iii) the Rudenko–Soluyan similarity solution when $\delta=-\frac{1}{2}$ .