Hydrodynamic chains and the classification of their Poisson brackets

Abstract

Infinite component Poisson brackets of the Dubrovin-Novikov type [Sov. Math. Dokl. 27, 665–669 (1983)] 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 [Am. Math. Soc. Transl. 170, 38–58 (1995)] type can be simplified in the Liouville coordinates. The Jacobi identity for infinite component Poisson brackets of the Ferapontov-Mokhov type [Russ. Math. Surveys 45, 218–219 (1990)] is presented in the Liouville coordinates