Integrable string and hydrodynamical type models and nonlocal brackets

The closed string model in the background gravity field is considered as a bi-Hamiltonian system in assumption that string model is the integrable model for particular kind of the background fields. The dual nonlocal Poisson brackets(PB), depending of the background fields and of their derivatives, are obtained. The integrability condition is formulated as the compatibility of the bi-Hamiltonity condition and the Jacobi identity of the dual PB. It is shown that the dual brackets and dual Hamiltonians can be obtained from the canonical PB and from the initial Hamiltonian by imposing the second kind constraints on the initial dynamical system, on the closed string model in the constant background fields, as example. The hydrodynamical type equation was obtained. Two types of the nonlocal brackets are introduced. Constant curvature and time-dependent metrics are considered. It is shown that the Jacobi identities for the nonlocal brackets have particular solution for the space-time coordinates, as matrix representation of the simple Lie group.