  The one-dimensional Euler-Poisson system arises in the study of phenomena of
plasma such as plasma solitons, plasma sheaths, and double layers. When the
system is rescaled by the Gardner-Morikawa transformation, the rescaled system
is known to be formally approximated by the Korteweg-de Vries (KdV) equation.
In light of this, we show existence of solitary wave solutions of the
Euler-Poisson system in the stretched moving frame given by the transformation,
and prove that they converge to the solitary wave solution of the associated
KdV equation as the small amplitude parameter tends to zero. Our results assert
that the formal expansion for the rescaled system is mathematically valid in
the presence of solitary waves and justify Sagdeev's formal approximation for
the solitary wave solutions of the pressureless Euler-Poisson system. Our work
extends to the isothermal case.
