Lie-Group Approach to Perturbative Renormalization Group Method

Goto, Shin-itiro; Masutomi, Yuji; Nozaki, Kazuhiro

doi:10.1143/ptp.102.471

Cited by 36 publications

(48 citation statements)

References 6 publications

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“…In the context of Hamiltonian systems subject to small-amplitude Hamiltonian perturbations, it has been shown [47,48] that the RG method yields results equivalent to those obtained from canonical Hamiltonian perturbation theory, up to and including O( 2 ). Finally, for completeness, we note that RG has also been applied to derive reduced or amplitude equations for certain nonlinear partial differential equations; see [4][5][6]12,13,10,24,28]. …”

Section: Relation Of This Analysis To Other Results About the Rg Methmentioning

confidence: 99%

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Deville¹,

Harkin²,

Holzer³

et al. 2008

Physica D: Nonlinear Phenomena

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For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502-4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré-Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré-Birkhoff normal forms for these systems up to and including terms of O( 2 ), where is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O(1/ ).

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Section: Relation Of This Analysis To Other Results About the Rg Methmentioning

confidence: 99%

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Deville¹,

Harkin²,

Holzer³

et al. 2008

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

show abstract

“…After their works, many studies of the RG method have been done [10][11][12][13]16,21,22,[26][27][28][29][30][31][32]35,36,42,45,46,48,49,53,[55][56][57][58][59]. Kunihiro [28,29] interpreted an approximate solution obtained by the RG method as the envelope of a family of regular perturbation solutions.…”

Section: Introductionmentioning

confidence: 99%

Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

Chiba¹

2009

SIAM J. Appl. Dyn. Syst.

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The renormalization group (RG) method is one of the singular perturbation methods which is used in search for asymptotic behavior of solutions of differential equations. In this article, time-independent vector fields and time (almost) periodic vector fields are considered. Theorems on error estimates for approximate solutions, existence of approximate invariant manifolds and their stability, inheritance of symmetries from those for the original equation to those for the RG equation, are proved. Further it is proved that the RG method unifies traditional singular perturbation methods, such as the averaging method, the multiple time scale method, the (hyper-) normal forms theory, the center manifold reduction, the geometric singular perturbation method and the phase reduction. A necessary and sufficient condition for the convergence of the infinite order RG equation is also investigated.

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“…In this section, we introduce the renormalization group method [12][13][14][15][16] using a FRW cosmological model. To obtain temporal evolution of the solution of the non-linear differential equation, we usually apply a perturbative expansion.…”

Section: Renormalization Group Methodsmentioning

confidence: 99%

“…In this paper, we investigate the cosmological back reaction problem using the second order cosmological perturbation and the renormalization group method [12][13][14][15][16][17][18][19]. We treat a dust dominated universe and examine how the inhomogeneity affect the evolution of the background FRW universe.…”

Section: Introductionmentioning

confidence: 99%

Renormalization group approach to cosmological back reaction problems

Nambu

2000

Phys. Rev. D

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We investigated the back reaction of cosmological perturbations on the evolution of the universe using the second order perturbation of the Einstein's equation. To incorporate the back reaction effect due to the inhomogeneity into the framework of the cosmological perturbation, we used the renormalization group method. The second order zero mode solution which appears by the non-linearities of the Einstein's equation is regarded as a secular term of the perturbative expansion, we renormalized a constant of integration contained in the background solution and absorbed the secular term to this constant. For a dust dominated universe, using the second order gauge invariant quantity, we derived the renormalization group equation which determines the effective dynamics of the Friedman-Robertson-Walker universe with the back reaction effect in a gauge invariant manner. We obtained the solution of the renormalization group equation and found that perturbations of the scalar mode and the long wavelength tensor mode works as positive spatial curvature, and the short wavelength tensor mode as radiation fluid. PACS number(s): 04

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Lie-Group Approach to Perturbative Renormalization Group Method

Cited by 36 publications

References 6 publications

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

Renormalization group approach to cosmological back reaction problems

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