Geometrical characteristics associated with stability and bifurcations of periodic travelling waves in reaction-diffusion systems. (English) Zbl 0602.35052
The paper deals with the vectorial reaction-diffusion equation
\[
(1)\quad u_ t=du_{xx}+f(u);\quad -\infty <x<+\infty,
\]
where D is a diagonal diffusivity matrix. When D and f satisfy appropriate conditions, the equation (1) has a one-parameter family of periodic travelling wave solutions of the form
\[
(2)\quad u=\psi_ 0[x+c(\ell)t;\ell],
\]
where \(\ell\) is the spatial period and c(\(\ell)\) is the propagation speed of the wave. By means of a sequence of characteristics of the \(\ell\)- periodic wave given by (2), an interesting analysis of the stability and bifurcations of travelling waves is established.
Reviewer: P.Renno
MSC:
35K55 | Nonlinear parabolic equations |
35K57 | Reaction-diffusion equations |
35A30 | Geometric theory, characteristics, transformations in context of PDEs |
35B10 | Periodic solutions to PDEs |
35B35 | Stability in context of PDEs |
35B32 | Bifurcations in context of PDEs |
35B40 | Asymptotic behavior of solutions to PDEs |