Dubrovin-Novikov type Poisson brackets (DN-brackets). (English. Russian original) Zbl 0671.58006
Funct. Anal. Appl. 22, No. 4, 336-338 (1988); translation from Funkts. Anal. Prilozh. 22, No. 4, 92-93 (1988).
The paper is concerned with canonical forms of the Poisson bracket
\[
\{u^ i(x),u^ j(y)\}=\sum g^{ij\alpha}(u(x))\partial \delta (x- y)/\partial x^{\alpha}+\sum (\partial u^ k(x\quad)/\partial x^{\alpha})b_ k^{ij\alpha}(u(x))\delta (x-y)
\]
of hydrodynamical type (where \(g^{ij\alpha}(u)\), \(u(u^ 1,...,u^ N)\), are nondegenerate metric tensors, \(b_ k^{ij\alpha}=\sum g^{is\alpha} \Gamma^{j\alpha}_{sk}\), and the symmetric connections \(\Gamma^{j\alpha}_{ik}\) related to the metrics are of zero curvature) correcting some previous results presented in several other articles.
Reviewer: J.Chrastina
References:
[1] | B. A. Dubrovin and S. P. Novikov, Dokl. Akad. Nauk SSSR,270, No. 4, 781-785 (1983). |
[2] | B. A. Dubrovin and S. P. Novikov, Dokl. Akad. Nauk SSSR,279, No. 2, 294-297 (1984). |
[3] | S. P. Novikov, Usp. Mat. Nauk,40, No. 4, 79-89 (1985). |
[4] | I. M. Gel’fand and I. Ya. Dorfman, Funkts. Anal. Prilozhen.,13, No. 4, 13-30 (1979). |
[5] | I. M. Gel’fand and I. Ya. Dorfman, Funkts. Anal. Prilozhen.,15, No. 3, 23-40 (1981). |
[6] | A. A. Balinskii and S. P. Novikov, Dokl. Akad. Nauk SSSR,283, No. 5, 1036-1039 (1985). |
[7] | E. I. Zel’manov, Dokl. Akad. Nauk SSSR,292, No. 6, 1294-1297 (1987). |
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