L. Schimanskygeier et al., ANALYSIS OF DOMAIN-SOLUTIONS IN REACTION-DIFFUSION SYSTEMS, Zeitschrift fur Physik. B, Condensed matter, 96(3), 1995, pp. 417-427
We investigate a standard model for bistable reaction-diffusion-system
s, which shares characteristic properties with the van-der-Pol oscilla
tor for distributed generators and the FitzHugh-Nagumo system. In this
system we study the effect of a long ranging inhibitor. As a main res
ult we show the existence of two inhomogeneous stationary solutions -
the smaller one is always a saddle which corresponds to a critical nuc
leus, while the larger one arises as a stable solution. In carrying ou
t the linear stability analysis for these solutions, we have to treat
the Schrodinger-equation for a double-well potential. This is done app
roximately by a supersymmetric approach which yields the eigenvalues a
nd eigenfunctions of the Schrodinger-equation. Furthermore we compare
our analytical findings with numerical results - especially the occurr
ence of oscillating solutions is shown.