Reductions of Kinetic Equations to Finite Component Systems

We consider two distinguish approaches for extraction of finite component systems from kinetic equations. The first method is based on the theory of generalized functions, which in simplest case is nothing but the so called multi flow hydrodynamics well known in plasma physics. An alternative is the so called the moment decomposition method successfully utilized for hydrodynamic chains. The method of hydrodynamic reductions successfully utilized in the theory of integrable hydrodynamic chains is applied to the local and nonlocal kinetic equations. N component reductions parameterized by N?1 arbitrary constants for non-hydrodynamic chain arising in the theory of high frequency nonlinear waves in electron plasma are found. These evolution dispersive systems equipped by a local Hamiltonian structure possess periodic solutions.