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

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish
Pages (from-to)167-174
Number of pages8
JournalDynamics and Stability of Systems
Volume13
Issue number2
Publication statusPublished - Jun 1998

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traveling waves
reaction-diffusion equations

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

In: Dynamics and Stability of Systems, Vol. 13, No. 2, 06.1998, p. 167-174.

Research output: Contribution to journalArticle

TY - JOUR

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

AU - Sherratt, Jonathan A.

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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.

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