Acta mathematica scientia,Series A ›› 1998, Vol. 18 ›› Issue (S1): 1-12.

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Explicit Wavefront Solutions of a Reaction-diffusion Equation and Global Analysis of a Planar Cubic System

Kang Dongsheng   

  1. Zhumadian Teacher's College, Henan 463000
  • Received:1996-03-11 Online:1998-12-26 Published:1998-12-26

Abstract:

It this paper,author studies the traveiling-wave solutions of the reaction-diffusion e-quation
ut=uxx+u(1-u)(u-a)(1)where 0 < a < 1,and c ≥ 0.Four types of traveiling-wave solutions are obtained with theirminimal or unique wave speeds c. Explicit form of some wave-front solutions including theHuxley wave are also established. The results are then applied to investigate the following pla-nar cubic system ζ=η, η=++(1+a)ξ2+ζ3.(2)It is well known that in the (U,U') phaes plane,the equation (1)is equivalent to the system(2). H. P. Mckean studied (1) by means of the traveiling-wave solution tnethod,and acquiredsome phase portraits for a∈(0,1/2) and c ≥ 0.Their results in this paper,however,glve allthe phase portraits for system (2) together with the bifurcation diagram.

Key words: Reaction-diffusion equation, traveiling-wave solution, explicit wave-front solution, wave sneed, planar nolynomlalsvstem, global phase portrait, bifurcation diagram

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