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Ibragimov, Nail H.; Kovalev, V. F.; Pustovalov, V. V.

doi:10.1023/a:1015061100660

Cited by 34 publications

(15 citation statements)

References 11 publications

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“…Based on this result ( 9)-( 11) for E 1 , the second step in [1] includes the equations E 2 (5) and E 3 (6) into the symmetry analysis, however, not generally, but rather with the following specifically chosen ansatz for the infinitesimals (equations (A.35)- (A.38)…”

Section: Summary Of the Key Results Obtained In [1]mentioning

confidence: 99%

“…When including equation E 4 (7), however, the situation is different: besides the trivial restriction (equation (53) in [1]) 6 dωb 0 (t, x, ω) = 0, (28)…”

Section: Summary Of the Key Results Obtained In [1]mentioning

confidence: 99%

“…The solution constraint (32) says that the infinitesimal ξ 3 , given by (31), should not depend on the spatial coordinates x 1 and x 2 . Sure, at first glance (31) and (32) stand in conflict with each 6 Note that the explicit form of the restrictions (28) and (29) can also be represented differently, for example, when splitting the arbitrary function C(ω) additively into two separate ones C(ω) = C 1 (ω) + C 2 (ω), as has been done in [1]. For instance, if C 2 is linked to b 0 and C 1 to ξ 3 , then (28) and (29) can also be equivalently written as dωb 0 (t, x, ω) = dω C 2 (ω) and ξ 3 3 = 6c 11 (x) + C 1 (ω), respectively.…”

Section: First Perspective: the Smoothness-axiom Of Lie-groupsmentioning

confidence: 99%

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Comment on ‘Conformal invariance of the Lundgren–Monin–Novikov equations for vorticity fields in 2D turbulence’

Frewer,

Khujadze

2021

J. Phys. A: Math. Theor.

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The recent claim by Grebenev et al (2017 J. Phys. A: Math. Theor. 50 435502) that the inviscid 2D Lundgren–Monin–Novikov (LMN) equations on a zero-vorticity characteristic naturally would reveal local conformal invariance when only analyzing these by means of a classical Lie-group symmetry approach, is invalid. To note is that within this comment the (possible) existence of conformal invariance in 2D turbulence is not questioned, only the conclusion as given by Grebenev et al and their approach how this invariance was derived is criticized and refuted herein. In fact, the algebraic derivation of conformal invariance of the 2D LMN vorticity equations in Grebenev et al (2017 J. Phys. A: Math. Theor. 50 435502) is flawed. Its correction will reveal a globally constant scaling in the fields, instead of a local one as claimed.

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Section: Summary Of the Key Results Obtained In [1]mentioning

confidence: 99%

“…When including equation E 4 (7), however, the situation is different: besides the trivial restriction (equation (53) in [1]) 6 dωb 0 (t, x, ω) = 0, (28)…”

Section: Summary Of the Key Results Obtained In [1]mentioning

confidence: 99%

Section: First Perspective: the Smoothness-axiom Of Lie-groupsmentioning

confidence: 99%

See 1 more Smart Citation

Comment on ‘Conformal invariance of the Lundgren–Monin–Novikov equations for vorticity fields in 2D turbulence’

Frewer,

Khujadze

2021

J. Phys. A: Math. Theor.

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“…The κ J variable is related to κ by means of an integral relation [49] (for brevity, we write only one argument of the coordinate of the generator, the one with respect to which the integration is performed):…”

Section: The Rg Symmetry Of Functionals In the Hopf Equationmentioning

confidence: 99%

Renormalization-group symmetries for solutions of nonlinear boundary value problems

Kovalev

Ширков

2008

Phys.-Usp.

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Approximately 10 years ago, the method of renormalization-group symmetries entered the field of boundary value problems of classical mathematical physics, stemming from the concepts of functional self-similarity and of the Bogoliubov renormalization group treated as a Lie group of continuous transformations. Overwhelmingly dominating practical quantum field theory calculations, the renormalization-group method formed the basis for the discovery of the asymptotic freedom of strong nuclear interactions and underlies the Grand Unification scenario. This paper describes the logical framework of a new algorithm based on the modern theory of transformation groups and presents the most interesting results of application of the method to differential and/or integral equation problems and to problems that involve linear functionals of solutions. Examples from nonlinear optics, kinetic theory, and plasma dynamics are given, where new analytical solutions obtained with this algorithm have allowed describing the singularity structure for self-focusing of a laser beam in a nonlinear medium, studying generation of harmonics in weakly inhomogeneous plasma, and investigating the energy spectra of accelerated ions in expanding plasma bunches.

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“…From this perspective the symmetry analysis of these quantum field theories could be mapped to the symmetry analysis of the integro-differential equations of motion, at least in the classical approximation. The Lie algebra approach to symmetries of integro-differential equations is not new, and there is a quite extensive literature available on this subject [12][13][14]. However, in contrast to local field theories, the methods and strategies of analysis strongly vary from case to case, and no universal method is known yet.…”

Section: Introductionmentioning

confidence: 99%