Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

Authors: J. A. Leach and E. Hanaç
Journal: Quart. Appl. Math. 74 (2016), 337-359
MSC (2010): Primary 35B40, 35K57
Published electronically: March 16, 2016
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Abstract: In this paper, we consider an initial-value problem for the Burgers-Fisher equation

$\displaystyle u_{t}+ k u u_{x}=u_{xx}+u(1-u), \quad -\infty <x<\infty , \quad t>0,$

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

$\displaystyle c^{*}(k)=\left \{ \begin {array}{ll} 2, \quad & -\infty < k \le 2, \\ \frac {2}{k}+\frac {k}{2}, & 2<k <\infty . \end{array} \right .$    

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 )$.

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Additional Information

J. A. Leach
Affiliation: School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
Email: j.a.leach@bham.ac.uk

E. Hanaç
Affiliation: School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
Email: hana.esen@gmail.com

DOI: https://doi.org/10.1090/qam/1421
Received by editor(s): September 30, 2014
Published electronically: March 16, 2016
Article copyright: © Copyright 2016 Brown University

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