Issues

 / 

1997

 / 

November

  

Reviews of topical problems


Hamiltonian formalism for nonlinear waves

,
Landau Institute for Theoretical Physics, Russian Academy of Sciences, ul. Kosygina 2, Moscow, 119334, Russian Federation

The Hamiltonian description of hydrodynamic type systems in application to plasmas, hydrodynamics, and magnetohydrodynamics is reviewed with emphasis on the problem of introducing canonical variables. The relation to other Hamiltonian approaches, in particular natural-variable Poisson brackets, is pointed out. It is shown that the degeneracy of noncanonical Poisson brackets relates to a special type of symmetry, the relabeling transformations of fluid-particle Lagrangian markers, from which all known vorticity conservation theorems, such as Ertel’s, Cauchy’s, Kelvin’s, as well as vorticity frozenness and the topological Hopf invariant, are derived. The application of canonical variables to collisionless plasma kinetics is described. The Hamiltonian structure of Benney’s equations and of the Rossby wave equation is discussed. Davey-Stewartson’s equation is given the Hamiltonian form. A general method for treating weakly nonlinear waves is presented based on classical perturbation theory and the Hamiltonian reduction technique.

Fulltext pdf (411 KB)
Fulltext is also available at DOI: 10.1070/PU1997v040n11ABEH000304
PACS: 52.30.−q, 52.35.Ra, 52.55.Fa (all)
DOI: 10.1070/PU1997v040n11ABEH000304
URL: https://ufn.ru/en/articles/1997/11/a/
000071302300001
Citation: Zakharov V E, Kuznetsov E A "Hamiltonian formalism for nonlinear waves" Phys. Usp. 40 1087–1116 (1997)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Оригинал: Захаров В Е, Кузнецов Е А «Гамильтоновский формализм для нелинейных волн» УФН 167 1137–1167 (1997); DOI: 10.3367/UFNr.0167.199711a.1137

References (85) Cited by (273) Similar articles (20) ↓

  1. M.I. Shliomis “Magnetic fluidsSov. Phys. Usp. 17 153–169 (1974)
  2. V.V. Yan’kov “Attractors and frozen-in invariants in turbulent plasmaPhys. Usp. 40 477–493 (1997)
  3. V.S. Edel’man “Levitated electronsSov. Phys. Usp. 23 227–244 (1980)
  4. A.P. Protogenov “Anyon superconductivity in strongly-correlated spin systemsSov. Phys. Usp. 35 (7) 535–571 (1992)
  5. V.E. Zakharov, E.A. Kuznetsov “Solitons and collapses: two evolution scenarios of nonlinear wave systemsPhys. Usp. 55 535–556 (2012)
  6. A.A. Vedenov, E.P. Velikhov, R.Z. Sagdeev “STABILITY OF PLASMASov. Phys. Usp. 4 332–369 (1961)
  7. V.F. Kop’ev, S.A. Chernyshev “Vortex ring oscillations, the development of turbulence in vortex rings and generation of soundPhys. Usp. 43 663–690 (2000)
  8. F.V. Dolzhanskii, V.A. Krymov, D.Yu. Manin “Stability and vortex structures of quasi-two-dimensional shear flowsSov. Phys. Usp. 33 (7) 495–520 (1990)
  9. V.F. Kovalenko, E.L. Nagaev “Photoinduced magnetismSov. Phys. Usp. 29 297–321 (1986)
  10. V.B. Shikin “Instability and reconstruction of a charged liquid surfacePhys. Usp. 54 1203–1225 (2011)
  11. G.A. Smolenskii, R.V. Pisarev, I.G. Sinii “Birefringence of light in magnetically ordered crystalsSov. Phys. Usp. 18 410–429 (1975)
  12. A.A. Chernyshov, K.V. Karelsky, A.S. Petrosyan “Subgrid-scale modeling for the study of compressible magnetohydrodynamic turbulence in space plasmasPhys. Usp. 57 421–452 (2014)
  13. Yu.A. Stepanyants, A.L. Fabrikant “Propagation of waves in hydrodynamic shear flowsSov. Phys. Usp. 32 783–805 (1989)
  14. P.K. Shukla, B. Eliasson “Nonlinear aspects of quantum plasma physicsPhys. Usp. 53 51–76 (2010)
  15. K.P. Zybin, V.A. Sirota “Stretching vortex filaments model and the grounds of statistical theory of turbulencePhys. Usp. 58 556–573 (2015)
  16. A.M. Bykov, I.N. Toptygin “Particle kinetics in highly turbulent plasmas (renormalization and self-consistent field methods)Phys. Usp. 36 (11) 1020–1052 (1993)
  17. A.A. Vlasov “The vibrational properties of an electron gasSov. Phys. Usp. 10 721–733 (1968)
  18. B.E. Meierovich “Toward the realization of electromagnetic collapseSov. Phys. Usp. 29 506–529 (1986)
  19. V.P. Silin “Absorption of radiation by turbulent laser plasmasSov. Phys. Usp. 28 136–152 (1985)
  20. V.P. Budaev, S.P. Savin, L.M. Zelenyi “Investigation of intermittency and generalized self-similarity of turbulent boundary layers in laboratory and magnetospheric plasmas: towards a quantitative definition of plasma transport featuresPhys. Usp. 54 875–918 (2011)

The list is formed automatically.

© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions