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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.

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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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Îðèãèíàë: Çàõàðîâ Â Å, Êóçíåöîâ Å À «Ãàìèëüòîíîâñêèé ôîðìàëèçì äëÿ íåëèíåéíûõ âîëí» ÓÔÍ 167 1137–1167 (1997); DOI: 10.3367/UFNr.0167.199711a.1137

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