Renormalization-group symmetries for solutions of nonlinear boundary value problems

V.F. Kovalev^a, D.V. Shirkov^b
^aInstitute of Mathematical Modelling, Russian Academy of Sciences, Miusskya pl. 4a, Moscow, 125047, Russian Federation
^bJoint Institute for Nuclear Research, 6 Joliot-Curie Str., Dubna, Moscow Region, 141980, Russian Federation

About 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 draws on lectures delivered at the XIII School for Nonlinear Waves, Nizhnii Novgorod, Russia, 1 — 7 March 2006 [see V F Kovalev, D V Shirkov “Renormalization group symmetry for solutions of boundary value problems” in Nonlinear Waves 2006 (Ed. by A V Gaponov-Grekhov) (N. Novgorod: IAP RAS, 2007) p. 433] to describe the logical framework of a new algorithm based on the modern theory of transformation groups and to present 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 analytic 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.

PACS: 02.30.Jr, 11.10.Hi, 42.65.−k (all)
DOI: 10.1070/PU2008v051n08ABEH006590
URL: https://ufn.ru/en/articles/2008/8/c/
Web of Science

000261856600003
Scopus

2-s2.0-57549113455
ADS NASA

2008PhyU...51..815K
Citation: Kovalev V F, Shirkov D V "Renormalization-group symmetries for solutions of nonlinear boundary value problems" Phys. Usp. 51 815–830 (2008)

BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Ковалев В Ф, Ширков Д В «Ренормгрупповые симметрии для решений нелинейных краевых задач» УФН 178 849–865 (2008); DOI: 10.3367/UFNr.0178.200808d.0849

References (60) Cited by (27) Similar articles (20) ↓

O.V. Rudenko “Nonlinear dynamics of quadratically cubic systems” Phys. Usp. 56 683–690 (2013)
A.V. Belinsky, M.Kh. Shulman “Quantum nature of a nonlinear beam splitter” Phys. Usp. 57 1022–1034 (2014)
A.V. Kukushkin “A technique for solving the wave equation and prospects for physical applications arising therefrom” Phys. Usp. 36 (2) 81–93 (1993)
V.S. Zapasskii “On electromagnetically induced transparency in the degenerate Λ-scheme” Phys. Usp. 52 179–181 (2009)
V.P. Silin, P.V. Silin “Bifurcation properties of the bremsstrahlung harmonics generated by a pumping field in plasmas” Phys. Usp. 50 729–740 (2007)
S.V. Sazonov “Superluminal electromagnetic solitons in nonequilibrium media” Phys. Usp. 44 631–644 (2001)
V.I. Ritus “Finite value of the bare charge and the relation of the fine structure constant ratio for physical and bare charges to zero-point oscillations of the electromagnetic field in a vacuum” Phys. Usp. 65 468–486 (2022)
V.P. Bykov “Basic properties of squeezed light” Sov. Phys. Usp. 34 (10) 910–924 (1991)
B.Ya. Zel’dovich, M.J. Soileau “Bi-frequency pendulum on a rotary platform: modeling various optical phenomena” Phys. Usp. 47 1239–1255 (2004)
A.N. Oraevskii “Superluminal waves in amplifying media” Phys. Usp. 41 1199–1209 (1998)
S.V. Goupalov “Classical problems with the theory of elasticity and the quantum theory of angular momentum” Phys. Usp. 63 57–65 (2020)
V.P. Bykov, V.I. Tatarskii “Perturbation theory for resolvents as applied to problems in radiation theory” Sov. Phys. Usp. 34 (2) 167–184 (1991)
V.G. Niz’ev “Dipole-wave theory of electromagnetic diffraction” Phys. Usp. 45 553–559 (2002)
V.S. Pronskikh “Measurement problems: contemporary discussions and models” Phys. Usp. 63 192–200 (2020)
E.A. Nelin “Impedance model for quantum-mechanical barrier problems” Phys. Usp. 50 293–299 (2007)
P.S. Landa, D.I. Trubetskov, V.A. Gusev “Delusions versus reality in some physics problems: theory and experiment” Phys. Usp. 52 235–255 (2009)
G.I. Broman, O.V. Rudenko “Submerged Landau jet: exact solutions, their meaning and application” Phys. Usp. 53 91–98 (2010)
D.V. Karlovets “On dual representation in classical electrodynamics” Phys. Usp. 53 817–824 (2010)
V.B. Braginskii “Adolescent years of experimental physics” Phys. Usp. 46 81–87 (2003)
A.V. Belinskii, A.S. Chirkin “Bernstein’s paradox of entangled quantum states” Phys. Usp. 56 1126–1131 (2013)

The list is formed automatically.