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Hamiltonian formalism for nonlinear waves

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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.

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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)
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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 (267) Similar articles (20)

  1. Zakharov V E, Kuznetsov E A Sov. Sci. Rev. (Ed. S P Novikov) 91 1310 (1986)
  2. Zakharov V E, L’vov V S Izv. Vyssh. Uchebn. Zaved. Radiofiz. 18 1470 (1975)
  3. Zakharov V E Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17 431 (1974)
  4. Zakharov V E, Musher S L, Rubenchik A M Phys. Rep. 129 285 (1985)
  5. Whitham G B Linear and Nonlinear Waves (Chichester, Sussex, England: Wiley, 1974)
  6. Kuznetsov E A, Rubenchik A M, Zakharov V E Phys. Rep. 142 103 (1986)
  7. Holm D D et al. Phys. Rep. 123 1 (1985)
  8. Arnol’d V I Dokl. Akad. Nauk SSSR 162 975 (1965)
  9. Arnol’d V I Usp. Mat. Nauk 24 (3) 225 (1969)
  10. Arnol’d V I Matematicheskie Metody Klassicheskoi Mekhaniki (Mathematical Methods of Classical Mechanics, Moscow: Nauka, 1974) [Translated into English (New York: Springer Verlag, 1989)]
  11. Arnold V I, Kozlov V V, Neishtadt A I, in Itogi Nauki i Tekhniki Ser. Sovremennye Problemy Matematiki Vol. 3 (Moscow: VINITI, 1985)
  12. Landau L D Zh. Eksp. Teor. Fiz. 11 592 (1941)
  13. Dzyaloshinskii I E, Volovick G E Ann. Phys. 125 67 (1980)
  14. Novikov S P Sov. Sci. Rev. 91 1 (1986)
  15. Morrison P J, Greene J M Phys. Rev. Lett. 45 790 (1980)
  16. Morrison P J Phys. Lett. A 80 383 (1980)
  17. Lamb H Hydrodynamics (Cambridge: Cambridge Univ. Press, 1932)
  18. Kuznetsov E A, Mikhailov A V Phys. Lett. A 77 37 (1980)
  19. Bateman H Proc. R. Soc. London Ser. A 125 598 (1929); Partial Differential Equations of Mathematical Physics (Cambridge: Cambridge Univ. Press, 1944)
  20. Davydov D I Dokl. Akad. Nauk SSSR 69 165 (1949)
  21. Khalatnikov I M Zh. Eksp. Teor. Fiz. 23169 (1952)
  22. Seliger R L, Whitham G B Proc. R. Soc. London Ser. A 305 1 (1968)
  23. Kontorovich V M, Kravchik H, Time V "Hamiltonian description of nonpotential motion in the presence of a free surface in ordinary and magnetohydrodynamdics" Preprint 3-158, IRE, Acad. Sci, Ukr. SSR (Khar’kov, 1980)
  24. Zakharov V E, Filonenko N N Dokl. Akad. Nauk USSR 170 1292 (1966)
  25. Zakharov V E Zh. Prikl. Mekh. Tekh. Fiz. (2) 86 (1968)
  26. Voronovich A G Izv. Akad. Nauk SSSR Ser. Fiz. Atmosfer i Okeana 15 82 (1979)
  27. Goncharov V P Izv. Akad. Nauk SSSR Ser. Fiz. Atmosfer i Okeana 16 473 (1980)
  28. Miropol’skii Yu Z Dinamika Vnutrennikh Gravitatsionnykh Voln v Okeane (Dynamics of Internal Gravitational Waves in the Ocean, Leningrad: Gidrometeoizdat, 1981)
  29. Goncharov V P, Pavlov V I Problemy Gidrodinamiki v Gamil’tonovom Opisanii (Problems of Hydrodynamics in the Hamiltonian Description, Moscow: Moscow University Press, 1993)
  30. Zakharov V E Physica D 3 193 (1981)
  31. Zakharov V E Funk. Anal. Prilozh. 14 (2) 15 (1980)
  32. Kupershmidt B A, Manin Yu I Funk. Anal. Prilozh. 11 (3) 31 (1977); 12 (1) 25 (1978)
  33. Zakharov V E, Kuznetsov E A Dokl. Akad. Nauk USSR 194 1288 (1970) [Sov. Phys. Dokl. 15 913 (1971)]
  34. Zakharov V E Zh. Eksp. Teor. Fiz. 60 1714 (1971) [Sov. Phys. JETP 33 927 (1971)]
  35. Kuznetsov E A Zh. Eksp. Teor. Fiz. 62 584 (1972) [Sov. Phys. JETP 35 310 (1972)]
  36. Kirillov A A Elementy Teorii Predstavlenii (Elements of the Theory of Representations, Moscow: Nauka, 1978) [Translated into English (Berlin, New York: Springer Verlag, 1978)]
  37. Kostant B, Proc. U.S.-Japan Seminar on Differential Geometry (Kyoto 1965, Tokyo, 1966)
  38. Zakharov V E, Monin A S, Piterbarg L I Dokl. Akad. Nauk SSSR 295 1061 (1987) [Sov. Phys. Dokl. 32 626 (1987)]; Zakharov V E, Piterbarg L I Dokl. Akad. Nauk SSSR 295 86 (1987) [Sov. Phys. Dokl. 32 560 (1987)]
  39. Piterbarg L I Phys. Lett. A 205 149 (1996)
  40. Zakharov V E et al. Teoriya Solitonov (Theory of Solitons, Moscow: Nauka, 1980)
  41. Salmon R, in Conf. Proc. Am. Inst. Phys. (88) 127 (1981)
  42. Ertel H Meteorol. Z. 59 277 (1942)
  43. Salmon R Ann. Rev. Fluid Mech. 20 225 (1988)
  44. Eckart C Phys. Rev. 54 920 (1938); Phys. Fluids 3 421 (1960)
  45. Eckart C Phys Fluids 6 1037 (1963)
  46. Newcomb W A, in Proc. Symp. Appl. Math. 18 152 (1967)
  47. Zakharov V E, Kuznetsov E A "Hamiltonian formalism for systems of hydropdynamic type" Preprint No. 186, Int. Automation and Electrometry, Sib. Branch USSR Ac. Sci. (Novosibirsk, 1982)
  48. Padhye N, Morrison P J Phys. Lett. A 219 287 (1996)
  49. Boozer A H Magnetic Field Line Hamiltonian Princeton Plasma Phys. Lab. Rep. No. PPPLR-2094R
  50. Lin C C, in Liquid Helium, Proceedings of the Enrico Fermi International School of Physics, Course XXI (New York: Academic, 1963)
  51. Yan’kov V V Pis’ma Zh. Eksp. Teor. Fiz. 58 516 (1993) [JETP Lett. 58 520 (1993)]; Yankov V V Zh. Eksp. Teor. Fiz. 107 414 (1995) [JETP 80 234 (1995)]
  52. Uby L, Isichenko M B, Yankov V V Phys. Rev. E 52 932 (1995)
  53. van Saarloos W Physica A 108 557 (1981)
  54. Landau L D, Lifshits E M Gidrodinamika (Hydrodynamics, Moscow: Nauka, 1988)
  55. Landau L D, Lifshits E M Mekhanika (Mechanics, Moscow: Nauka, 1965)
  56. Moreau J J C.R. Acad. Sci. 252 2810 (1961)
  57. Moffatt H K J. Fluid Mech. 35 117 (1969)
  58. Faddeev L D Lett. Math. Phys. 1 289 (1976)
  59. Dubrovin B A, Novikov S P, Fomenko A T Sovremennaya Geometriya (Modern Geometry, Moscow: Nauka, 1979) [Translated into English (New York: Springer Verlag, 1992)]
  60. Whitehead J H C Proc. Natl. Acad. Sci. U.S.A. 33 (5) 115 (1947)
  61. Volovik G E, Mineev V P Zh. Eksp. Teor. Fiz. 72 2256 (1977) [Sov. Phys. JETP 45 1186 (1977)]
  62. Abrashkin A A, Zen’kovich D A, Yakubovich E I Izv. Vyssh. Uchebn. Zaved. Radiofiz. XXXIX 783 (1996)
  63. Abarbanel H D I et al. Philos. Trans. R. Soc. London Ser. A 318 349 (1986)
  64. Lewis D, Marsden J, Montgomery R, Ratiu T Physica D 18 391 (1986)
  65. Ebin D, Marsden J Ann. Math. 92 102 (1970)
  66. Karpman V I Nelineinye Volny v Dispergiruyushchikh Sredakh (Nonlinear Waves in Dispersive Media, Moscow: Nauka, 1973)
  67. Kuznetsov E A, Spector M D, Zakharov V E Phys. Rev. E 49 1283 (1994)
  68. Kuznetsov E A, Spector M D Zh. Eksp. Teor. Fiz. 71 262 (1976) [Sov. Phys. JETP 44 136 (1976)]
  69. Rouhi A, Wright J Phys. Rev. E 48 1850 (1993)
  70. Dirac P A M Proc. R. Soc. London Ser. A 212 330 (1952)
  71. Frenkel A, Levich E, Stilman L Phys. Lett. A 88 461 (1982)
  72. Gordin V A, Petviashvili V I Fiz. Plazmy 13 509 (1987)
  73. Henyey F, in Conf. Proc. Am. Inst. Phys. (88) 85 (1982)
  74. Moiseev S S et al. Zh. Eksp. Teor. Fiz. 83 215 (1982) [Sov. Phys. JETP 56 117 (1982)]
  75. Gavrilin B L, Zaslavskii M M Dokl. Akad. Nauk SSSR 192 48 (1970)
  76. Virasoro M A Phys. Rev. Lett. 47 1181 (1981)
  77. Bloembergen N Nonlinear Optics (Ed. W A Benjamin, Mass.: Reading, 1965)
  78. Bogolyubov N N, Shirkov D V Vvedenie v Teoriyu Kvantovykh Polei (Introduction to the Theory of Quantized Fields, Moscow: Nauka, 1976)
  79. Zakharov V E, Rubenchik A M Zh. Prikl. Mekh. Tekh. Fiz. (5) 84 (1972)
  80. Davey A, Stewartson K Proc. R. Soc. Ser. A 338 101 (1974)
  81. Goncharov V P Dokl. Akad. Nauk SSSR 313 27 (1990)
  82. Zakharov V E Funk. Anal. Prilozh. 23 (3) 24 (1989)
  83. Krasitskii V P Zh. Eksp. Teor. Fiz. 98 1644 (1990) [Sov. Phys. JETP 71 921 (1990)]
  84. Zakharov V E Osnovy Fiziki Plazmy (Handbook of Plasma Physics, Eds A Galeeev, R Sudan, New York: Elsevier, 1984)
  85. Kadomtsev B B, Petviashvili V I Dokl. Akad. Hauk SSSR 192 753 (1970) [Sov. Phys. Dokl. 15 539 (1970)]
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