Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 103, 54 pages      arXiv:1203.5732      http://dx.doi.org/10.3842/SIGMA.2012.103
Contribution to the Special Issue “Geometrical Methods in Mathematical Physics”

Whitham's Method and Dubrovin-Novikov Bracket in Single-Phase and Multiphase Cases

Andrei Ya. Maltsev
L.D. Landau Institute for Theoretical Physics, 1A Ak. Semenova Ave., Chernogolovka, Moscow reg., 142432, Russia

Received April 23, 2012, in final form December 11, 2012; Published online December 24, 2012

Abstract
In this paper we examine in detail the procedure of averaging of the local field-theoretic Poisson brackets proposed by B.A. Dubrovin and S.P. Novikov for the method of Whitham. The main attention is paid to the questions of justification and the conditions of applicability of the Dubrovin-Novikov procedure. Separate consideration is given to special features of single-phase and multiphase cases. In particular, one of the main results is the insensitivity of the procedure of bracket averaging to the appearance of ''resonances'' which can arise in the multi-phase situation.

Key words: quasiperiodic solutions; slow modulations; Hamiltonian structures.

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