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 SIGMA, 2012, Volume 8, 103, 54 pages (Mi sigma780)

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

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.

Keywords: quasiperiodic solutions; slow modulations; Hamiltonian structures

DOI: https://doi.org/10.3842/SIGMA.2012.103

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ArXiv: 1203.5732
MSC: 37K05; 35B10; 35B15; 35B34; 35L65
Received: April 23, 2012; in final form December 11, 2012; Published online December 24, 2012
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Citation: Andrei Ya. Maltsev, “Whitham's Method and Dubrovin–Novikov Bracket in Single-Phase and Multiphase Cases”, SIGMA, 8 (2012), 103, 54 pp.

Citation in format AMSBIB
\Bibitem{Mal12} \by Andrei~Ya.~Maltsev \paper Whitham's Method and Dubrovin--Novikov Bracket in Single-Phase and Multiphase Cases \jour SIGMA \yr 2012 \vol 8 \papernumber 103 \totalpages 54 \mathnet{http://mi.mathnet.ru/sigma780} \crossref{https://doi.org/10.3842/SIGMA.2012.103} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000312648800001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84871789756} 

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This publication is cited in the following articles:
1. Maltsev A.Ya., “The Multi-Dimensional Hamiltonian Structures in the Whitham Method”, J. Math. Phys., 54:5 (2013), 053507
2. Maltsev A.Ya., “on the Minimal Set of Conservation Laws and the Hamiltonian Structure of the Whitham Equations”, J. Math. Phys., 56:2 (2015), 023510
3. Maltsev A.Ya., “On the canonical forms of the multi-dimensional averaged Poisson brackets”, J. Math. Phys., 57:5 (2016), 053501
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