  The known nonlinear kinetic equations (in particular, the wave kinetic
equation and the quantum Nordheim -- Uehling -- Uhlenbeck equations) are
considered as a natural generalization of the classical spatially homogeneous
Boltzmann equation. To this goal we introduce the general Boltzmann - type
kinetic equation that depends on a function of four real variables $ F(x, y; v,
w) $. The function $ F $ is assumed to satisfy certain commutation relations.
The general properties of this equation are studied. It is shown that the above
mentioned kinetic equations correspond to different forms of the function
(polynomial) $ F $. Then the problem of discretization of the general Boltzmann
- type kinetic equation is considered on the basis of ideas which are similar
to those used for construction of discrete models of the Boltzmann equation.
The main attention is paid to discrete models of the wave kinetic equation. It
is shown that such models have a monotone functional similar to Boltzmann $ H
$-function. The existence and uniqueness theorem for global in time solution of
the Cauchy problem for these models is proved. Moreover it is proved that the
solution converges to the equilibrium solution when time goes to infinity. The
properties of the equilibrium solution and the connection with solutions of the
wave kinetic equation are discussed. The problem of approximation of the
Boltzmann-type equation by its discrete models and the problem of construction
of normal discrete models are also discussed. The paper contains a concise
introduction to the Boltzmann equation and its main properties. In principle,
it allows to read the paper without any preliminary knowledge in kinetic
theory.
