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23 February 2020
  » arxiv » hep-th/9905088

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Renormalization-group Method for Reduction of Evolution Equations; invariant manifolds and envelopes
S.-I. Ei ; K. Fujii ; T. Kunihiro ;
Date 13 May 1999
Journal Annals Phys. 280 (2000) 236-298
Subject High Energy Physics - Theory; Mathematical Physics; Pattern Formation and Solitons; Exactly Solvable and Integrable Systems | hep-th cond-mat gr-qc hep-ph math-ph math.MP nlin.PS nlin.SI patt-sol solv-int
AbstractThe renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set $t_0=t$ is naturally understood where $t_0$ is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator $A$ in the evolution equation has no Jordan cell; when $A$ has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be the Polchinski’s theorem for renormalizable field theories. We apply the method to interface dynamics such as kink-anti-kink and soliton-soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.
Source arXiv, hep-th/9905088
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