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We study the interaction of two counter-propagating electromagnetic waves invacuum in the Born-Infeld electrodynamics. First we investigate the Born casefor linearly polarized beams, ${\bf E}\cdot{\bf B}=0$, i. e. $\mathfrak{G}^2=0$(crossed field configuration), which is identical for Born-Infeld and Bornelectrodynamics; subsequently we study the general Born--Infeld case for beamswhich are nonlinearly polarized, $\mathfrak{G}^2\neq0$.  In both cases, we show that the nonlinear field equations decouple usingself-similar solutions and investigate the shock wave formation. We show thatthe only nonlinear solutions are exceptional travelling wave solutions whichpropagate with constant speed and which do not turn into shocks.  In the Born case, we naturally obtain exceptional wave solutions forcounter-propagating (real photon-photon scattering) and for a co-propagating(non-interacting) beam orientation we investigate their direction ofpropagation. In the Born--Infeld case, we have additionally chosen thesolutions which have constant phase velocities to match the limits of phasevelocities of the background field in the Born case. We obtain two types ofexceptional wave solutions, then we numerically analyze which phase velocitiescorrespond to the counter- or co-propagating beams and subsequently wedetermine the direction of propagation of the exceptional waves.

On the absence of shock waves and vacuum birefringence in Born-Infeld electrodynamics