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

In this article, we consider an initial-value problem for the generalized Burgers' equation. The normalized Burgers' equation considered is given by u(t) + t(delta) uu(x) = u(xx), -infinity < x < infinity, t > 0, where -1/2 <= delta not equal 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 >= 0 and u(x, 0) = u(-) for x < 0, where u(+) and u(-) are problem parameters with u(+) not equal 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 > -1/2 and u(+) > u(-), ( ii) a Taylor shock (hyperbolic tangent) profile when delta > -1/2 and u(+) < u(-) and ( iii) the Rudenko-Soluyan similarity solution when delta = -1/2.