  We discuss the theory of spin waves in non-degenerate ultra-cold gases, and
compare various methods which can be used to obtain appropriate kinetic
equations. We then study non-hydrodynamic situations, where the amplitude of
spin waves is sufficiently large to bring the system far from local
equilibrium. In the first part of the article, we compare two general methods
which can be used to derive a kinetic equation for a dilute gas of atoms
(bosons or fermions) with two internal states (treated as a pseudo-spin 1/2).
The collisional methods are in the spirit of Boltzmann's original derivation of
his kinetic equation where, at each point of space, the effects of all sorts of
possible binary collisions are added. We discuss two different versions of
collisional methods, the Yvon-Snider approach and the S matrix approach. The
second method uses the notion of mean field, which modifies the drift term of
the kinetic equation, in the line of the Landau theory of transport in quantum
liquids. For a dilute cold gas, it turns out that all these derivations lead to
the same drift terms in the transport equation, but differ in the precise
expression of the collision integral and in higher order gradient terms. In the
second part of the article, the kinetic equation is applied to spin waves in
trapped ultra-cold gases. Numerical simulations are used to illustrate the
strongly non-hydrodynamic character of the spin waves recently observed with
trapped Rb87 atoms. The decay of the phenomenon, which takes place when the
system relaxes back towards equilibrium, is also discussed, with a short
comment on decoherence.
