### Abstract

The Fisher-KPP equation u_{t} = u_{xx} + u(1 - u) has a travelling wave solution for all speeds = 2. Initial data that decrease monotonically from 1 to 0 on - 8 < x < 8, with u(x, 0) = O_{s}(e^{-?x}) as x ? 8, are known to evolve to a travelling wave, whose speed depends on ?. Here, it is shown that the relationship between wave speed and ? can be recovered by linearizing the Fisher-KPP equation about u = 0 and explicitly solving the linear equation. Moreover, the calculation predicts that in the case ? > 1, the solution for u_{x}/u itself evolves to a transition wave, moving ahead of the (minimum speed) u wave at the greater speed of 2?. Behind this transition, u_{x}/u = -x/(2t), while ahead of it, u_{x}/u = -?. The paper goes on to discuss the potential applications of the method to systems of coupled reaction-diffusion equations.

Original language | English |
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Pages (from-to) | 167-174 |

Number of pages | 8 |

Journal | Dynamics and Stability of Systems |

Volume | 13 |

Issue number | 2 |

Publication status | Published - Jun 1998 |

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*Dynamics and Stability of Systems*, vol. 13, no. 2, pp. 167-174.

**On the transition from initial data to travelling waves in the Fisher-KPP equation.** / Sherratt, Jonathan A.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the transition from initial data to travelling waves in the Fisher-KPP equation

AU - Sherratt, Jonathan A.

PY - 1998/6

Y1 - 1998/6

N2 - The Fisher-KPP equation ut = uxx + u(1 - u) has a travelling wave solution for all speeds = 2. Initial data that decrease monotonically from 1 to 0 on - 8 < x < 8, with u(x, 0) = Os(e-?x) as x ? 8, are known to evolve to a travelling wave, whose speed depends on ?. Here, it is shown that the relationship between wave speed and ? can be recovered by linearizing the Fisher-KPP equation about u = 0 and explicitly solving the linear equation. Moreover, the calculation predicts that in the case ? > 1, the solution for ux/u itself evolves to a transition wave, moving ahead of the (minimum speed) u wave at the greater speed of 2?. Behind this transition, ux/u = -x/(2t), while ahead of it, ux/u = -?. The paper goes on to discuss the potential applications of the method to systems of coupled reaction-diffusion equations.

AB - The Fisher-KPP equation ut = uxx + u(1 - u) has a travelling wave solution for all speeds = 2. Initial data that decrease monotonically from 1 to 0 on - 8 < x < 8, with u(x, 0) = Os(e-?x) as x ? 8, are known to evolve to a travelling wave, whose speed depends on ?. Here, it is shown that the relationship between wave speed and ? can be recovered by linearizing the Fisher-KPP equation about u = 0 and explicitly solving the linear equation. Moreover, the calculation predicts that in the case ? > 1, the solution for ux/u itself evolves to a transition wave, moving ahead of the (minimum speed) u wave at the greater speed of 2?. Behind this transition, ux/u = -x/(2t), while ahead of it, ux/u = -?. The paper goes on to discuss the potential applications of the method to systems of coupled reaction-diffusion equations.

UR - http://www.scopus.com/inward/record.url?scp=0032096980&partnerID=8YFLogxK

M3 - Article

VL - 13

SP - 167

EP - 174

JO - Dynamics and Stability of Systems

JF - Dynamics and Stability of Systems

SN - 0268-1110

IS - 2

ER -