Derivation of Amplitude Equations

A proper formulation in the perturbative renormalization group method is presented to deduce amplitude equations. The formulation makes it possible not only avoiding a serious diiculty in the previous reduction to amplitude equations by eliminating all of the secular terms but also consistent derivation of higher-order correction to amplitude equations. Recently a novel method based on the perturbative renormalization group (RG) theory has been developed as an asymptotic singular perturbation technique by L.Y.Chen, N.Goldenfeld and Y.Oono 1]. Their renormalization group method (RGM) removes secular or divergent terms from a perturbation series by renormalizing integral constants of lower order solutions. Although they have impressively succeeded in application to ordinary diier-ential equations 2], a consistent formulation of the RG method does not have been established for application to partial diierential equations. In fact, we have recently pointed out 7],,8] that all of secular solutions of a perturbation series does not have been eliminated yet in the previous derivation of amplitude equations from partial diierential equations 2],,3]. In this paper, we present a suitable formulation of a perturbation problem in RGM in order to avoid such a diiculty and deduce some amplitude equations and their higher-order correction consistently in the framework of RGM. A crucial part of our formulation is to scale all variables except one independent variable so that the non-scaled variable is taken as a parameter of RG. Therefore, the renormalization group equations (RGE) or amplitude 1