  We study two counter-propagating electromagnetic waves in the vacuum within
the framework of the Heisenberg-Euler formalism in quantum electrodynamics. We
show that the non-linear field equations decouple for ordinary wave case and
can be solved exactly. We solve the non-linear field equations assuming the
solution in a form of a Riemann wave. We discuss the properties of the
nonlinear electromagnetic wave propagating in the quantum vacuum, such as the
wave steepening, subsequent generation of high order harmonics and
electromagnetic shock wave formation with electron-positron pair generation at
the shock wave front.
