Issues

 / 

2008

 / 

August

  

Methodological notes


Renormalization-group symmetries for solutions of nonlinear boundary value problems

 a,  b
a Institute of Mathematical Modelling, Russian Academy of Sciences, Miusskya pl. 4a, Moscow, 125047, Russian Federation
b Joint 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.

Fulltext is available at IOP
PACS: 02.30.Jr, 11.10.Hi, 42.65.−k (all)
DOI: 10.1070/PU2008v051n08ABEH006590
URL: https://ufn.ru/en/articles/2008/8/c/
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 (23) Similar articles (20) ↓

  1. O.V. Rudenko “Nonlinear dynamics of quadratically cubic systems56 683–690 (2013)
  2. A.V. Belinsky, M.Kh. Shulman “Quantum nature of a nonlinear beam splitter57 1022–1034 (2014)
  3. A.N. Oraevskii “Superluminal waves in amplifying media41 1199–1209 (1998)
  4. A.V. Kukushkin “A technique for solving the wave equation and prospects for physical applications arising therefrom36 (2) 81–93 (1993)
  5. V.S. Zapasskii “On electromagnetically induced transparency in the degenerate Λ-scheme52 179–181 (2009)
  6. V.P. Silin, P.V. Silin “Bifurcation properties of the bremsstrahlung harmonics generated by a pumping field in plasmas50 729–740 (2007)
  7. B.Ya. Zel’dovich, M.J. Soileau “Bi-frequency pendulum on a rotary platform: modeling various optical phenomena47 1239–1255 (2004)
  8. S.V. Sazonov “Superluminal electromagnetic solitons in nonequilibrium media44 631–644 (2001)
  9. V.P. Bykov “Basic properties of squeezed light34 (10) 910–924 (1991)
  10. V.G. Niz’ev “Dipole-wave theory of electromagnetic diffraction45 553–559 (2002)
  11. P.S. Landa, V.F. Marchenko “On the linear theory of waves in media with periodic structures34 (9) 830–834 (1991)
  12. S.P. Efimov “Fock theory modification into 4-d coordinate space. Harmonic tensors in quantum Coulomb problem”, accepted
  13. P.S. Landa, D.I. Trubetskov, V.A. Gusev “Delusions versus reality in some physics problems: theory and experiment52 235–255 (2009)
  14. G.I. Broman, O.V. Rudenko “Submerged Landau jet: exact solutions, their meaning and application53 91–98 (2010)
  15. S.V. Vladimirov, Yu.O. Tyshetskiy “On description of a collisionless quantum plasma54 1243–1256 (2011)
  16. A.V. Burenin “Symmetry of quantum intramolecular dynamics45 753–776 (2002)
  17. M.V. Kuzelev, A.A. Rukhadze “On the quantum description of the linear kinetics of a collisionless plasma42 603–605 (1999)
  18. A.A. Abrashkin, E.N. Pelinovsky “On the relation between Stokes drift and the Gerstner wave61 307–312 (2018)
  19. G.A. Martynov “Nonequilibrium statistical mechanics, transport equations, and the second law of thermodynamics39 1045–1070 (1996)
  20. E.A. Andryushin, V.L. Ginzburg, A.P. Silin “Boundary conditions in the macroscopic theory of superconductivity36 (9) 854–857 (1993)

The list is formed automatically.

© 1918–2022 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions