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

:   ,    « » 178 849–865 (2008); DOI: 10.3367/UFNr.0178.200808d.0849

References (60) Cited by (22) ↓ Similar articles (20)

  1. Kovalev V F JNMP 18 163 (2021)
  2. Bychenkov V Yu, Kovalev V F Radiophys Quantum El 63 742 (2021)
  3. Kovalev V F, Bychenkov V Yu Phys. Rev. E 99 (4) (2019)
  4. Metelskii I I, Kovalev V F, Bychenkov V Yu Physics Of Plasmas 26 113113 (2019)
  5. Kovalev V F, Bychenkov V Yu Jetp Lett. 107 458 (2018)
  6. Konyukhov V K, Stepanov E V, Borisov S K Laser Phys. 28 055602 (2018)
  7. Kovalev V I Laser Phys. Lett. 14 015401 (2017)
  8. Metelskii I I, Kovalev V F, Bychenkov V Yu Plasma Phys. Rep. 43 175 (2017)
  9. Metelskii I I, Kovalev V F, Bychenkov V Yu Bull. Lebedev Phys. Inst. 43 16 (2016)
  10. Kovalev V I Frontiers in Optics 2016, (2016) p. FF1H.7
  11. Chirkunov Yu A J. Phys. A: Math. Theor. 48 395501 (2015)
  12. Bychenkov V Yu, Brantov A V et al Uspekhi Fizicheskikh Nauk 185 77 (2015) [Bychenkov V Yu, Brantov A V et al Phys.-Usp. 58 71 (2015)]
  13. Chirkunov Yu A Journal Of Mathematical Physics 56 101502 (2015)
  14. Kovalev V F, Bychenkov V Yu J. Exp. Theor. Phys. 121 1 (2015)
  15. Chirkunov Yu A, Nazarenko S V et al J. Phys. A: Math. Theor. 47 185501 (2014)
  16. Kovalev V F, Rudenko O V Acoust. Phys. 58 269 (2012)
  17. Kovalev V F, Popov K I, Bychenkov V Yu J. Exp. Theor. Phys. 114 25 (2012)
  18. Mitri F G Ultrasonics 51 496 (2011)
  19. Banerjee D, Bhattacharjee Ja K American Journal Of Physics 78 142 (2010)
  20. Banerjee D, Bhattacharjee Ja K J. Phys. A: Math. Theor. 43 062001 (2010)
  21. Rogovtsov N N Light Scattering Reviews 5 Chapter 7 (2010) p. 249
  22. Grigoriev Yu N, Ibragimov N H et al Lecture Notes In Physics Vol. Symmetries of Integro-Differential EquationsPlasma Kinetic Theory: VlasovMaxwell andRelated Equations806 Chapter 4 (2010) p. 145

© 1918–2021 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions