Hamiltonian formalism for nonlinear waves

V. E. Zakharov, E. A. Kuznetsov

    Research output: Contribution to journalReview articlepeer-review

    214 Citations (Scopus)

    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
    Pages (from-to)1087-1116
    Number of pages30
    JournalPhysics-Uspekhi
    Volume40
    Issue number11
    DOIs
    Publication statusPublished - Nov 1997

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