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 Izv. RAN. Ser. Mat., 1999, Volume 63, Issue 6, Pages 117–146 (Mi izv269)

Conservation of Hamiltonian structures in Whitham's averaging method

A. Ya. Maltsev

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: In this paper we consider Whitham's averaging method for systems with a local field-theoretic Hamiltonian structure. We prove that such a Hamiltonian structure is conserved in the process. The study is based on the procedure (suggested by Dubrovin and Novikov) of averaging the local Poisson bracket, for which we establish the necessary properties of conservation of the Jacobi identity and invariance.

DOI: https://doi.org/10.4213/im269

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English version:
Izvestiya: Mathematics, 1999, 63:6, 1171–1201

Bibliographic databases:

MSC: 58F05

Citation: A. Ya. Maltsev, “Conservation of Hamiltonian structures in Whitham's averaging method”, Izv. RAN. Ser. Mat., 63:6 (1999), 117–146; Izv. Math., 63:6 (1999), 1171–1201

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv269
• https://doi.org/10.4213/im269
• http://mi.mathnet.ru/eng/izv/v63/i6/p117

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. O. I. Mokhov, “Compatible Poisson Structures of Hydrodynamic Type and Associativity Equations”, Proc. Steklov Inst. Math., 225 (1999), 269–284
2. Maltsev A.Y., Novikov S.P., “On the local systems Hamiltonian in the weakly non-local Poisson brackets”, Physica D, 156:1–2 (2001), 53–80
3. A Ya Maltsev, “Weakly nonlocal symplectic structures, Whitham method and weakly nonlocal symplectic structures of hydrodynamic type”, J Phys A Math Gen, 38:3 (2005), 637
4. Liu S.Q., Zhang Y.J., “Deformations of semisimple bihamiltonian structures of hydrodynamic type”, Journal of Geometry and Physics, 54:4 (2005), 427–453
5. A. Ya. Maltsev, “Whitham systems and deformations”, J Math Phys (N Y ), 47:7 (2006), 073505
6. A Ya Maltsev, “The conservation of the Hamiltonian structures in the deformations of the Whitham systems”, J Phys A Math Theor, 43:6 (2010), 065202
7. V. L. Vereshchagin, “Single-Phase Averaging for the Ablowitz–Ladik Chain”, Math. Notes, 87:6 (2010), 797–806
8. Andrei Ya. Maltsev, “Whitham's Method and Dubrovin–Novikov Bracket in Single-Phase and Multiphase Cases”, SIGMA, 8 (2012), 103, 54 pp.
9. A. Ya. Maltsev, “On the minimal set of conservation laws and the Hamiltonian structure of the Whitham equations”, J. Math. Phys, 56:2 (2015), 023510
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