ON THE DEGENERATED SOFT-MODE INSTABILITY

We consider instabilities of a single mode with finite wavenumber in i nversion symmetric spatially one-dimensional systems, where the charac ter of the bifurcation changes from sub- to supercritical behaviour. S tarting from a general equation of motion the full amplitude equation is derived systematically and formulae for the dependence of the coeff icients on the system parameters are obtained. We emphasize the import ance of nonlinear derivative terms in the amplitude equation for the b ehaviour in the vicinity of the bifurcation point. In particular the n umerical values of the corresponding coefficients determine the region of coexistence between the stable trivial solution and stable spatial ly periodic patterns. Our approach clearly shows that similar consider ations fail for the case of oscillatory instabilities.