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Journal of Mathematical Biology

, Volume 35, Issue 6, pp 713–728 | Cite as

Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations

  • Faustino Sánchez-Garduño
  • Philip K. Maini

Abstract.

 In this paper we study the existence of one-dimensional travelling wave solutions u(x, t)=φ(xct) for the non-linear degenerate (at u=0) reaction-diffusion equation u t =[D(u)u x ] x +g(u) where g is a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing the Allee effect. We use a dynamical systems approach to prove: 1. the global bifurcation of a heteroclinic cycle (two monotone stationary front solutions), for c=0, 2. The existence of a unique value c*>0 of c for which φ(xc*t) is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for cc*. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation.

Key words: Sharp fronts Degenerate diffusion Hamiltonian Bifurcation of heteroclinic trajectories 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Faustino Sánchez-Garduño
    • 2
  • Philip K. Maini
    • 1
  1. 1.Centre for Mathematical Biology, Mathematical Institute, University of Oxford, 24-29 St. Giles’, Oxford OX1 3LB, UKGB
  2. 2.Departamento de Matemáticas, Facultad de Ciencias, UNAM, Circuito Exterior, C.U., México 04510, D.F., MexicoMX

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