Mokhov, O. I. The classification of nonsingular multidimensional Dubrovin-Novikov brackets. (English) Zbl 1180.37088 Funct. Anal. Appl. 42, No. 1, 33-44 (2008); translation from Funkts. Anal. Prilozh. 42, No. 1, 39-52 (2008). The paper starts from the problem of classification of the multidimensional Poisson brackets of hydrodynamic type \[ \{u^i(x),u^j(y)\} = \sum_{\alpha=1}^n(g^{ij\alpha}(u(x))\delta_{\alpha}(x-y)+ b_k^{ij\alpha}(u(x))u^k_{\alpha}(x)\delta(x-y)) \] which are useful in the Hamiltonian theory of systems of hydrodynamic type. The requirement of bilinearity and the Leibniz identity for a bracket of the above form, are equivalent to the condition that on arbitrary functionals \(I\) and \(J\) on the space of fields \(u(x)\) the bracket has the form \[ \displaystyle{\{I,J\}=\sum_{\alpha=1}^n\int{{\delta I}\over{\delta u^i(x)}}\left(g^{ij\alpha}(u(x)){{d}\over{dx^{\alpha}}}+b_k^{ij\alpha}(u(x)) u^k_{\alpha}(x)\right){{\delta J}\over{\delta u^j(x)}}d^nx} \] The main result - the classification theorem - reads as followsIf one of the metrics \(g^{ij\alpha}(u)\) for a nondegenerate multidimensional Poisson bracket above forms singular pairs with all the remaining metrics of this bracket, then this Poisson bracket can be reduced to a constant form by a local change of coordinates. Reviewer: Vladimir Răsvan (Craiova) Cited in 16 Documents MSC: 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) 35F21 Hamilton-Jacobi equations 35L65 Hyperbolic conservation laws Keywords:multidimensional Poisson bracket of hydrodynamic type; Dubrovin-Novikov bracket PDFBibTeX XMLCite \textit{O. I. Mokhov}, Funct. Anal. Appl. 42, No. 1, 33--44 (2008; Zbl 1180.37088); translation from Funkts. Anal. Prilozh. 42, No. 1, 39--52 (2008) Full Text: DOI arXiv References: [1] B. A. Dubrovin and S. P. Novikov, ”On Poisson brackets of hydrodynamic type,” Dokl. Akad. Nauk SSSR, 279:2 (1984), 294–297; English transl.: Soviet Math. Dokl., 30 (1984), 651–654. · Zbl 0591.58012 [2] B. A. Dubrovin and S. P. Novikov, ”The Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method,” Dokl. Akad. Nauk SSSR, 270:4 (1983), 781–785; English transl.: Soviet Math. Dokl., 27 (1983), 665–669. · Zbl 0553.35011 [3] S. P. Tsarev, ”Geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method,” Izv. Akad. Nauk SSSR. Ser. Matem., 54:5 (1990), 1048–1068; English transl.: Math. USSR Izv., 54:5 (1990), 397–419. [4] B. A. Dubrovin and S. P. Novikov, ”Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory,” Uspekhi Mat. Nauk, 44:6 (1989), 29–98; English transl.: Russian Math. Surveys, 44:6 (1989), 35–124. · Zbl 0712.58032 [5] E. V. Ferapontov and K. R. Khusnutdinova, ”On the integrability of (2 + 1)-dimensional quasilinear systems,” Comm. Math. Phys., 248 (2004), 187–206. · Zbl 1070.37047 [6] O. I. Mokhov, ”Poisson brackets of Dubrovin-Novikov type (DN-brackets),” Funkts. Anal. Prilozhen., 22:4 (1988), 92–93; English transl.: Functional Anal. Appl., 22:4 (1988), 336–338. · Zbl 0671.58006 [7] A. A. Balinskii and S. P. Novikov, ”Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras,” Dokl. Akad. Nauk SSSR, 283:5 (1985), 1036–1039; English transl.: Soviet Math. Dokl., 32 (1985), 228–231. · Zbl 0606.58018 [8] O. I. Mokhov, ”Compatible and almost compatible pseudo-Riemannian metrics,” Funkts. Anal. Prilozhen., 35:2 (2001), 24–36; English transl.: Functional Anal. Appl., 35:2 (2001), 100–10; http://xxx.arxiv.org/abs/math/0005051. · Zbl 1005.53016 [9] O. I. Mokhov, ”Symplectic and Poisson structures on loop spaces of smooth manifolds and integrable systems,” Uspekhi Mat. Nauk, 53:3 (1998), 85–192; English transl.: Russian Math. Surveys, 53:3 (1998), 515–622. · Zbl 0937.53001 [10] I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, John Wiley & Sons, Chichester, 1993. · Zbl 0717.58026 [11] N. I. Grinberg, ”On Poisson brackets of hydrodynamic type with a degenerate metric,” Uspekhi Mat. Nauk, 40:4 (1985), 217–218; English transl.: Russian Math. Surveys, 40:4 (1985), 231–232. · Zbl 0611.58017 [12] O. I. Mokhov, ”Hamiltonian systems of hydrodynamic type and constant curvature metrics,” Phys. Lett. A, 166:3–4 (1992), 215–216. · Zbl 0817.35079 [13] I. M. Gelfand and I. Ya. Dorfman, ”Hamiltonian operators and infinite-dimensional Lie algebras,” Funkts. Anal. Prilozhen., 15:3 (1981), 23–40; English transl.: Functional Anal. Appl., 15:3 (1981), 173–187. [14] F. Magri, ”A simple model of the integrable Hamiltonian equation,” J. Math. Phys., 19:5 (1978), 1156–1162. · Zbl 0383.35065 [15] O. I. Mokhov, ”Compatible flat metrics,” J. Appl. Math., 2:7 (2002), 337–370; http://xxx.arxiv.org/abs/math/0201224. · Zbl 1008.37041 [16] B. Dubrovin, ”Geometry of 2D topological field theories,” in: Lecture Notes in Math., vol. 1620, 1996, 120–348. · Zbl 0841.58065 [17] O. I. Mokhov, ”Integrability of the equations for nonsingular pairs of compatible flat metrics,” Teor. Mat. Fiz., 130:2 (2002), 233–250; English transl.: Theor. Math. Phys., 130:2 (2002), 198–212; http://xxx.arxiv.org/abs/math/0005081. · Zbl 1029.37046 [18] O. I. Mokhov, ”Flat pencils of metrics and integrable reductions of Lamé’s equations,” Uspekhi Mat. Nauk, 56:2 (2001), 221–222; English transl.: Russian Math. Surveys, 56:2 (2001), 416–418. · Zbl 1006.53017 [19] E. V. Ferapontov, ”Compatible Poisson brackets of hydrodynamic type,” J. Phys. A: Math. Gen., 34 (2001), 2377–2388; http://xxx.arxiv.org/abs/math/0005221. · Zbl 1010.37044 [20] A. Nijenhuis, ”X n-forming sets of eigenvectors,” Indag. Math., 13:2 (1951), 200–212. · Zbl 0042.16001 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. 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