Hamiltonian formalism for nonlinear waves

V. E. Zakharov, E. A. Kuznetsov

Research output: Contribution to journalReview article

183 Scopus citations

Abstract

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 non-canonical 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.

Original languageEnglish (US)
Pages (from-to)1087-1116
Number of pages30
JournalPhysics-Uspekhi
Volume40
Issue number11
DOIs
StatePublished - Nov 1 1997
Externally publishedYes

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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