Stability analysis of fronts in a tristable reaction-diffusion system

A stability analysis is performed analytically for the tristable reaction-diffusion equation, in which a quintic reaction term is approximated by a piecewise linear function. We obtain growth rate equations for two basic types of propagating fronts, monotonous and nonmonotonous ones. Their solutions show that the monotonous front is stable whereas the nonmonotonous one is unstable. It is found that there are two values of the growth rate for the most dangerous modes (corresponding to the longest possible wavelengths), <InlineEquation ID="Equ1"> <EquationSource Format="TEX">$\omega=0$</EquationSource> </InlineEquation> and <InlineEquation ID="Equ2"> <EquationSource Format="TEX">$\omega > 0$</EquationSource> </InlineEquation>, for the monotonous front, so that at <InlineEquation ID="Equ3"> <EquationSource Format="TEX">$\omega=0$</EquationSource> </InlineEquation> the perturbation eigenfunction is positive whereas when <InlineEquation ID="Equ4"> <EquationSource Format="TEX">$\omega > 0$</EquationSource> </InlineEquation> it changes sign. It is also noted that the eigenvalue <InlineEquation ID="Equ5"> <EquationSource Format="TEX">$\omega=0$</EquationSource> </InlineEquation> becomes negative in an inhomogeneous system with a particular (stabilizing) inhomogeneity. Counting arguments for the number of eigenmodes of the linear stability operator are presented. Copyright Springer-Verlag Berlin/Heidelberg 2004