Renormalization Group Method and Reductive Perturbation Method

It is shown that the renormalization group method does not necessarily eliminate all secular terms in perturbation series to partial differential equations and a functional subspace of renormalizable secular solutions corresponds to a choice of scales of independent variables in the reductive perturbation method. Recently a novel method based on the perturbative renormalization group theory has been developed as an asymptotic singular perturbation technique by L.Y.Chen, N.Goldenfeld and Y.Oono [1] and the usefulness of the method has been amply demonstrated [2]. Their renormalization group method (the RG method) removes secular or divergent terms from a perturbation series by renormalizing integral constants of lower order solutions. It is a crucial procedure to obtain secular solutions explicitly by perturbative analysis, which is usually easy and clear for ordinary differential equations (ODE) and the method has made impressive success in application to ODE [2]. However, applying the RG method to partial differetial equations (PDE), it should be noted that all of secular solutions to PDE can not be obtained unless a functional space of secular solutions is specified. This point has not been discussed explicitly in previous application of the RG method to PDE. It may be natural to restrict a functional space of secular solutions to a family of polynomial-type functions of independent variables. We shall show some physical examples where all of secular solutions of polynomial-type can not be removed by the renormalization procedure and we must impose further restrictions on the functional space in order to remove secular solutions. Unless a functional space of secular solutions is specified, the RG method does not necessarily yield the unique renormalization group (RG) equation in application to PDE . The purpose of this letter is to show through some examples that a functional subspace of secular solutions corresponds to a choice of scales of independent variables in the reductive perturbation (RP) method [5].