Hamiltonian formalism for nonlinear waves

Hamiltonian description for nonlinear waves in plasma, hydrodynamics and magnetohydrodynamics is presented. The main attention is paid to the problem of canonical variables introducing. The connection with other approaches of the Hamiltonian structure introducing is presented, in particular, with the help of the Poisson brackets expressed in terms of natural variables. It is shown that the degeneracy of the noncanonical Poisson brackets is connected for the system of hydrodynamic type with the specific symmetry, namely, with the relabeling transformations of the Lagrangian markers of fluid particles. All known theorems about the vorticity conservation (the Ertel's, Cauchy's and Kelvin's theorems, the frosenness of vorticity and conservation of the topological Hopf invariant) are a sequence of this symmetry. The canonical variables are introduced into the collisionless plasma kinetics and into the Benney equation. The problem of Hamiltonian structures is discussed for surface and internal waves as well as for the Rossby waves. The Hamiltonian structure also is introduced for the Davey-Stuartson equation, describing the interaction of quasimonochromatic waves with the induced low-frequency medium motion. At the end of this survey a general method for investigation of weak nonlinear waves is considered, based on both the classical perturbation theory and reduction of Hamiltonians.