  We investigate the steady-state spin diffusion for ultracold spin-orbit
coupled $^{40}$K gas by the kinetic spin Bloch equation approach both
analytically and numerically. Four configurations, i.e., the spin diffusions
along two specific directions with the spin polarization perpendicular
(transverse configuration) and parallel (longitudinal configuration) to the
effective Zeeman field are studied. It is found that the behaviors of the
steady-state spin diffusion for the four configurations are very different,
which are determined by three characteristic lengths: the mean free path
$l_{\tau}$, the Zeeman oscillation length $l_{\Omega}$ and the spin-orbit
coupling oscillation length $l_{\alpha}$. It is analytically revealed and
numerically confirmed that by tuning the scattering strength, the system can be
divided into {\it five} regimes: I, weak scattering regime ($l_{\tau}\gtrsim
l_{\Omega}, l_{\alpha}$); II, Zeeman field-dominated moderate scattering regime
($l_{\Omega}\ll l_{\tau}\ll l_{\alpha}$); III, spin-orbit coupling-dominated
moderate scattering regime ($l_{\alpha}\ll l_{\tau}\ll l_{\Omega}$); IV,
relatively strong scattering regime ($l_{\tau}^c\ll l_{\tau}\ll l_{\Omega},
l_{\alpha}$); V, strong scattering regime ($l_{\tau}\ll l_{\Omega},
l_{\alpha},l_{\tau}^c$), with $l_{\tau}^c$ representing the crossover length
between the relatively strong and strong scattering regimes. In different
regimes, the behaviors of the spacial evolution of the steady-state spin
polarization are very rich, showing different dependencies on the scattering
strength, Zeeman field and spin-orbit coupling strength. The rich behaviors of
the spin diffusions in different regimes are hard to be understood in the
framework of the simple drift-diffusion model or the direct inhomogeneous
broadening picture in the literature. ...
