  Two fundamental models in plasma physics are given by the
Vlasov-Poisson-Landau system and the compressible Euler-Poisson system which
both capture the complex dynamics of plasmas under the self-consistent electric
field interactions at the kinetic and fluid levels, respectively. Although
there have been extensive studies on the long wave limit of the Euler-Poisson
system towards Korteweg-de Vries equations, few results on this topic are known
for the Vlasov-Poisson-Landau system due to the complexity of the system and
its underlying multiscale feature. In this article, we derive and justify the
Korteweg-de Vries equations from the Vlasov-Poisson-Landau system modelling the
motion of ions under the Maxwell-Boltzmann relation. Specifically, under the
Gardner-Morikawa transformation $$ (t,x,v)\to
(\delta^{\frac{3}{2}}t,\delta^{\frac{1}{2}}(x-\sqrt{\frac{8}{3}}t),v) $$ with $
\varepsilon^{\frac{2}{3}}\leq \delta\leq \varepsilon^{\frac{2}{5}}$ and
$\varepsilon>0$ being the Knudsen number, we construct smooth solutions of the
rescaled Vlasov-Poisson-Landau system over an arbitrary finite time interval
that can converge uniformly to smooth solutions of the Korteweg-de Vries
equations as $\delta\to 0$. Moreover, the explicit rate of convergence in
$\delta$ is also obtained. The proof is obtained by an appropriately chosen
scaling and the intricate weighted energy method through the micro-macro
decomposition around local Maxwellians.
