Pavlov, Maxim V. Hydrodynamic chains and the classification of their Poisson brackets. (English) Zbl 1112.37062 J. Math. Phys. 47, No. 12, 123514, 15 p. (2006). Summary: Infinite component Poisson brackets of Dubrovin-Novikov type are considered. The corresponding Jacobi identity is significantly simplified in the Liouville coordinates since the skew-symmetry condition is automatically satisfied. The concept of M Poisson bracket connected with hydrodynamic chains is introduced. Then the Jacobi identity is a nonlinear system of equations in partial derivatives which can be completely integrated. In such a case, a classification of infinite component Poisson brackets of the Dubrovin-Novikov type can be obtained. Two simplest examples, \(M=0\) and \(M=1\), are considered. Also infinite component Poisson brackets of the Ferapontov type can be simplified in the Liouville coordinates. The Jacobi identity for infinite component Poisson brackets of the Ferapontov-Mokhov type is presented in the Liouville coordinates Cited in 2 Documents MSC: 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35Q53 KdV equations (Korteweg-de Vries equations) PDFBibTeX XMLCite \textit{M. V. Pavlov}, J. Math. Phys. 47, No. 12, 123514, 15 p. (2006; Zbl 1112.37062) Full Text: DOI arXiv References: [1] Balinsky A. A., Dokl. Akad. Nauk SSSR 283 pp 1036– (1985) [2] Balinsky A. A., Sov. Math. Dokl. 32 pp 228– (1985) [3] Benney D. J., Stud. Appl. Math. 52 pp 45– (1973) · Zbl 0259.35011 [4] DOI: 10.1088/0305-4470/35/48/309 · Zbl 1040.37053 [5] DOI: 10.1088/0305-4470/35/48/309 · Zbl 1040.37053 [6] Dorfman, I. Ya.Dirac Structures and Integrability of Nonlinear Evolution Equations, Nonlinear Science: Theory and Applications (Wiley, New York, 1993), p. 176. · Zbl 0717.58026 [7] DOI: 10.1070/RM1989v044n06ABEH002300 · Zbl 0712.58032 [8] DOI: 10.1070/RM1989v044n06ABEH002300 · Zbl 0712.58032 [9] Ferapontov E. V., Am. Math. Soc. Transl. 170 pp 33– (1995) [10] DOI: 10.1088/0305-4470/37/8/007 · Zbl 1040.35042 [11] DOI: 10.1088/0305-4470/37/8/007 · Zbl 1040.35042 [12] DOI: 10.1063/1.2354590 · Zbl 1112.37056 [13] Ferapontov E. V., Russ. Math. Surveys 45 pp 218– (1990) [14] DOI: 10.1063/1.1542921 · Zbl 1061.37046 [15] DOI: 10.1070/RM1985v040n04ABEH003662 [16] Kupershmidt B. A., Proc. R. Ir. Acad., Sect. A 83 pp 45– (1983) [17] Kupershmidt, B. A. , Proceedings of the Berkeley-Ames Conference on Nonlinear Problems in Control and Fluid Dynamics, Berkeley, California, 1983 (Math Science, Brookline, MA, 1984), pp. 357–378. [18] DOI: 10.1007/BF00401929 · Zbl 0415.35068 [19] DOI: 10.1007/BF00401929 · Zbl 0415.35068 [20] DOI: 10.1007/BF00401929 · Zbl 0415.35068 [21] DOI: 10.1007/BF00401929 · Zbl 0415.35068 [22] DOI: 10.1016/S0167-2789(01)00280-9 · Zbl 0991.37041 [23] DOI: 10.1023/A:1021198811148 [24] DOI: 10.1023/A:1021198811148 [25] DOI: 10.1023/A:1024469316049 · Zbl 1043.37049 [26] DOI: 10.1023/A:1024469316049 · Zbl 1043.37049 [27] DOI: 10.1023/A:1024469316049 · Zbl 1043.37049 [28] Mokhov O. I., Teor. Mat. Fiz. 132 pp 60– (2002) [29] Mokhov O. I., Teor. Mat. Fiz. 132 pp 942– (2002) [30] DOI: 10.1007/978-1-4684-0274-2 [31] DOI: 10.1016/S0375-9601(98)00307-7 · Zbl 0947.35124 [32] Pavlov M. V., Theor. Math. Phys. 138 pp 55– (2004) [33] DOI: 10.1007/BF02468523 [34] DOI: 10.1007/BF02468523 [35] DOI: 10.1070/IM1991v037n02ABEH002069 · Zbl 0796.76014 [36] DOI: 10.1070/IM1991v037n02ABEH002069 · Zbl 0796.76014 [37] DOI: 10.1007/978-1-4615-2474-8_13 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.