2000/01/30 by Dmitrij V. Shirkov, Dmitrij V Shirkov, V. F. Kovalev +2 · 4 citations
Mathematics · Physics and Astronomy · #Advanced Fiber Laser Technologies #Laser-Matter Interactions and Applications #Nonlinear Photonic Systems #cond-mat.stat-mech #hep-th #math-ph #math.MP
paper · pdf · doi:10.1016/s0370-1573(01)00039-4
published as Phys.Rept. 352 (2001) 219-249 · Contribution to the proceedings of conference "RG 2000" (Taxco, Mexico, Jan. 1999). To be published in Physics Reports
arxiv created 2000/01/30 · crossref issued 2001/10/01 · crossref published 2001/10/01 · crossref published-print 2001/10/01 · openalex publication_date 2001/10/01 · crossref created 2002/07/25 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · crossref deposited 2020/01/13 · crossref indexed 2026/03/25 · openalex updated_date 2026/07/28
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.