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

We study the interaction of two counter-propagating electromagnetic waves in vacuum in the Born-Infeld electrodynamics. First we investigate the Born case for 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 Born electrodynamics; subsequently we study the general Born-Infeld case for beams which are nonlinearly polarized, $\mathfrak{G}^2\neq0$. In both cases, we show that the nonlinear field equations decouple using self-similar solutions and investigate the shock wave formation. We show that the only nonlinear solutions are exceptional travelling wave solutions which propagate with constant speed and which do not turn into shocks for our approximation. We obtain two types of exceptional wave solutions, then we numerically analyze which phase velocities correspond to the counter- or co-propagating beams and subsequently we determine the direction of propagation of the exceptional waves.