  In this paper we use a spin kinetic equation to study spin polarization
dynamics in 1D wires and 2D channels. This approach is valid in both diffusive
and ballistic spin transport regimes and, therefore, more general than the
usual spin drift-diffusion equations. In particular, we demonstrate that in
infinite 1D wires with Rashba spin-orbit interaction the exponential spin
relaxation decay can be modulated by an oscillating function. In the case of
spin relaxation in finite length 1D wires, it is shown that an initially
homogeneous spin polarization spontaneously transforms into a persistent spin
helix. An interesting sound waves echo-like behavior of initially localized
spin polarization packet is found in finite length wires. We show that a
propagating spin polarization profile reflects from a system boundary and
returns back to its initial position similarly to the reflectance of sound
waves from an obstacle. Green's function of spin kinetic equation is found for
both finite and infinite 1D systems. Moreover, we demonstrate explicitly that
the spin relaxation in 2D channels with Rashba and Dresselhaus spin-orbit
interactions of equal strength occurs similarly to that in 1D wires of finite
length. Finally, a simple transformation mapping 1D spin kinetic equation into
the Klein-Gordon equation with an imaginary mass is found thus establishing an
interesting connection between semiconductor spintronics and relativistic
quantum mechanics.
