  In this article, we will prove $L^2(\mathbb{R})$-stability of $1$-solitons
for the KdV equation by using exponential stability property of the semigroup
generated by the linearized operator. The proof follows the lines of recent
stability argument of Mizumachi [Asymptotic stability of lattice solitons in
the energy space, Comm. Math. Phys., (2009)] and Mizumachi, Pego and Quintero
[Asymptotic stability of solitary waves in the Benney-Luke model of water
waves, Differential Integral Equations, (2013)] which show stability in the
energy class by using strong linear stability of solitary waves in
exponentially weighted spaces. This gives an alternative proof of Merle and
Vega [$L^2$ stability of solitons for KdV equation, Int. Math. Res. Not.,
(2003)] which shows $L^2(\mathbb{R})$-stability of $1$-solitons for the KdV
equation by using the Miura transformation. Our argument is a refinement of
Pego and Weinstein [Asymptotic stability of solitary waves, Comm. Math. Phys.,
(1994)] that proves asymptotic stability of solitary waves in exponentially
weighted spaces. We slightly improve the $H^1$-stability of the modified KdV
equation as well.
