  We study two counter--propagating electromagnetic waves in the vacuum within
the framework of the Born--Infeld theory in quantum electrodynamics. By
choosing the crossed field case ${\bf E}\cdot{\bf B}=0$, i.e.
$\mathfrak{G}^2=0$, the Born--Infeld Lagrangian reduces to the Born Lagrangian,
therefore for this special case we present study which is identical for the
Born--Infeld and the Born electrodynamics. In this paper, we show that the
non--linear field equations decouple for ordinary wave case using self-similar
solutions and we investigate the shock wave steepening. We show that the only
solutions are exceptional traveling wave solutions which propagate with
constant speed and which do not turn into shocks. We discuss the phase shift
and the cross section of the process to be measured together with the our
proposed direct detection of the photon--photon scattering
\cite{KadlecovaKornBulanov2019,KadlecovaMine2019}. This work serves as a
preliminary study towards analyzing the waves in the more general case in the
Born--Infeld theory.
