  The Korteweg-de Vries (KdV) equation is known as a universal equation
describing various long waves in dispersive systems. In this article, we prove
that in a certain scaling regime, a large class of rough solutions to the
Boussinesq equation are approximated by the sums of two counter-propagating
waves solving the KdV equations. It extends the earlier result by
\cite{Schneider1998} to slightly more regular than $L^2$-solutions. Our proof
is based on robust Fourier analysis methods developed for the low regularity
theory of nonlinear dispersive equations.
