  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. $\mathfrak{G}^2=0$, we
present the solution of a statical problem which is identical for both the Born
and the Born-Infeld electrodynamics. The Born theory exhibits birefringence
effect contrary to the Born-Infeld theory, therefore we are able to investigate
shock wave development. In this paper, we show that the non-linear field
equations decouple for ordinary wave case using self-similar solutions and we
demonstrate the wave steepening, shock wave formation and higher-harmonic
generation. We discuss the phase shift and the cross section of the process.
This work serves as a preliminary study towards analyzing the shock waves in
the nonstatical case in the Born-Infeld theory.
