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A method to construct asymptotic solutions invariant under the renormalization group. (English) Zbl 1108.81040
Summary: There have been many studies concerning application of the renormalization group theory of particle physics as a singular perturbation method to treat differential equations since the work of the Illinois group. In this method, integral constants appearing in the lowest-order perturbed solution are renormalized in order to remove secular or divergent terms appearing in the naive perturbation solution and give a well-behaved asymptotic solution. We call this renormalization group method in singular perturbation theory the “conventional RG method” in this paper. A geometrical aspect of the conventional RG method has been studied by means of the envelopes of solutions. The conventional RG method and its variants depend more or less on the naive perturbation analysis. In connection to this, it is interesting that the geometrical symmetry of a differential equation with respect to continuous transformations, i.e., the Lie symmetry, is known to be useful for deriving new solutions from known solutions. There have been some pioneering works aimed of constructing the renormalization group method in terms of the Lie symmetry. However, to this time, there is no case in which have been derived asymptotic solutions in singular perturbation problems by means of the Lie symmetry. The purpose of this paper is to derive such asymptotic solutions in the framework of the Lie group and symmetry.

MSC:
81T17Renormalization group methods (quantum theory)
34C14Symmetries, invariants (ODE)
34C20Transformation and reduction of ODE and systems, normal forms
34E05Asymptotic expansions (ODE)
34E15Asymptotic singular perturbations, general theory (ODE)