Group analysis and renormgroup symmetries

Kovalev, V. F.; Pustovalov, V. V.; Shirkov, D. V.

doi:10.1063/1.532374

Cited by 43 publications

(55 citation statements)

References 32 publications

(71 reference statements)

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“…, l entering into a solution via the equations and/or boundary conditions. Adding parameters p to the list of independent variables z = {x, p} we treat BEs in this extended space as RM (31). Similarly, one can extend the space of differential variables by treating derivatives with respect to p as additional differential variables.…”

Section: Symmetry Of Solution In Mathematical Physicsmentioning

confidence: 99%

“…Differentiation with re-spect to these parameters gives additional DEs (embedding equations) that, together with BEs, form RM. In some cases, while calculating Lie point RGS, the role of embedding equations can be played by differential constraints (for details see [31]) that come from an invariance condition for BEs with respect to the Lie-Bäcklund 8 symmetry group.…”

Section: Symmetry Of Solution In Mathematical Physicsmentioning

confidence: 99%

“…Here X is extended 9 to all derivatives involved in F σ and the symbol | (31) means calculated on the frame (31). A system of linear homogeneous PDEs (34) for coordinates ξ i , η α , known as determining equations, is an overdetermined system as a rule.…”

Section: Symmetry Of Solution In Mathematical Physicsmentioning

confidence: 99%

“…Here, the procedure of RGS constructing makes use of the Lie-Bäcklund symmetry and is described as follows [31]. The manifold RM (step I) is defined by Eqs.…”

Section: Plane Geometrymentioning

confidence: 99%

See 3 more Smart Citations

The Bogoliubov renormalization group and solution symmetry in mathematical physics

Shirkov¹,

Kovalev²

2001

Physics Reports

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Evolution of the concept known in the theoretical physics as the Renormalization Group (RG) is presented. The corresponding symmetry, that has been first introduced in QFT in mid-fifties, is a continuous symmetry of a solution with respect to transformation involving parameters (e.g., of boundary condition) specifying some particular solution.After short detour into Wilson's discrete semi-group, we follow the expansion of QFT RG and argue that the underlying transformation, being considered as a reparameterisation one, is closely related to the self-similarity property. It can be treated as its generalization, the Functional Self-similarity (FS).Then, we review the essential progress during the last decade of the FS concept in application to boundary value problem formulated in terms of differential equations. A summary of a regular approach recently devised for discovering the RG = FS symmetries with the help of the modern Lie group analysis and some of its applications are given.As a main physical illustration, we give application of new approach to solution for a problem of self-focusing laser beam in a non-linear medium.

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Section: Symmetry Of Solution In Mathematical Physicsmentioning

confidence: 99%

Section: Symmetry Of Solution In Mathematical Physicsmentioning

confidence: 99%

Section: Symmetry Of Solution In Mathematical Physicsmentioning

confidence: 99%

“…Here, the procedure of RGS constructing makes use of the Lie-Bäcklund symmetry and is described as follows [31]. The manifold RM (step I) is defined by Eqs.…”

Section: Plane Geometrymentioning

confidence: 99%

See 2 more Smart Citations

The Bogoliubov renormalization group and solution symmetry in mathematical physics

Shirkov¹,

Kovalev²

2001

Physics Reports

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“…(2) Shirkov [3] clarified that the RG equation concerns with the initial values and emphasized the Lie group structure of the RG method [42]; he extracted the notion of functional selfsimilarity (FSS) as the essence of the exact RG. He claims that the Wilson RG is an approximation to the Bogoliubov RG which is exact [3].…”

Section: Introductionmentioning

confidence: 99%

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Ei¹,

Fujii²,

Kunihiro³

2000

Annals of Physics

145

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The 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 is semi-simple, i.e., diagonalizable; when A is not semi-simple and 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.

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The Bogoliubov Renormalization Group in Theoretical and Mathematical Physics

Shirkov¹

2000

Quantum Field Theory

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This text follows the line of a talk on Ringberg symposium dedicated to Wolfhart Zimmermann 70th birthday. The historical overview (Part 1) partially overlaps with corresponding text of my previous commemorative paper -see Ref.[61] in the list. At the same time second part includes some recent results in QFT (Sect. 2.1) and summarize (Sect. 2.4) an impressive progress of the "QFT renormalization group" application in mathematical physics.

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Group analysis and renormgroup symmetries

Cited by 43 publications

References 32 publications

The Bogoliubov renormalization group and solution symmetry in mathematical physics

The Bogoliubov renormalization group and solution symmetry in mathematical physics

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

The Bogoliubov Renormalization Group in Theoretical and Mathematical Physics

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