On the evolution of travelling wave solutions of the Burgers-Fisher equation

In this paper, we consider an initial-value problem for the Burgers-Fisher equation \[ u_{t}+ k u u_{x}=u_{xx}+u(1-u), \quad -\infty <x<\infty , \quad t>0,\] where $x$ and $t$ represent dimensionless distance and time respectively and $k (\not = 0)$ is a parameter. In particular, we consider the case when the initial data has a discontinuous compressive step, where $u(x,0)=1$ for $x \le 0$ and $u(x,0)=0$ for $x>0$. The method of matched asymptotic coordinate expansions is used to obtain the large-$t$ asymptotic structure of the solution to this problem, which exhibits the formation of a permanent form travelling wave propagating in the $+x$ direction with the minimum possible speed $c=c^*(k)$, where \begin{equation}\nonumber c^{*}(k)=\left \{ \begin {array}{ll} 2, \quad & -\infty < k \le 2, \\ \frac {2}{k}+\frac {k}{2}, & 2<k <\infty . \end{array} \right . \end{equation} The rate of convergence of the solution of the initial-value problem to the permanent form travelling wave is found to be algebraic in $t$, as $t \to \infty$, when $k \in (-\infty ,2]$ and exponential in $t$, as $t \to \infty$, when $k \in (2,\infty )$.