Weakly nonlinear analysis in spatially extended systems as a formal perturbation scheme

摘要

The well known concept, to reduce the spatio-temporal dynamics beyondinstabilities of trivial states to amplitude modulated patterns, is reviewedfrom the point of view of a formal perturbation expansion for generaldissipative partial differential equations. For codimension one instabilitiesclosed analytical formulas for all coefficients of the resulting amplitudeequation are given, with no further restriction on the basic equations ofmotion. Both the autonomous and the explicitly time-dependent case arediscussed. For the latter, the problem of strong resonances is addressedseparately. The formal character of the expansion allows for an analysis ofhigher-codimension instabilities like the Turing-Hopf instability and for thediscussion of principal limits of the amplitude approach in the present form.