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 nonlinear field equations decouple for ordinary wave case and can be solved exactly. We solve the nonlinear 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.