Bogolyubov renormalization group and symmetry of solution in mathematical physics

Shirkov, Dmitrij V. (Dubna, JINR) ; Kovalev, Vladimir F. (Moscow, Inst. Math. Modeling)

Published in: Phys.Rept.
Year: 2001
Vol.: 352
Page No: 219-249
Pages: 36

Abstract: 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.

Web-Page: http://www.adsabs.harvard.edu/abs/2001PhR...352..219S; http://alice.cern.ch/format/showfull?sysnb=2174083; http://www.jinr.ru/publish/Preprints/2000/e2-2000-9.pdf; http://www.sciencedirect.com/science?_ob=GatewayURL&_origin=SPIRES&_method=citationSearch&_volkey=03701573%23352%23219%23&_version=1&md5=d3b0f1e6e2db32725db380d14b6d5385; http://www1.jinr.ru/Preprints/2000/e2-2000-9.pdf; http://lt-jds.jinr.ru/record/23162/files/e2-2000-9.pdf

Note: Dedicated to the memory of Boris Medvedev

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