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

TL;DR

This paper investigates wave interactions in Born-Infeld electrodynamics, demonstrating that nonlinear solutions are limited to non-shock traveling waves, with no vacuum birefringence or shock wave formation.

Contribution

It provides a detailed analysis of wave interactions in Born-Infeld theory, revealing the absence of shock waves and vacuum birefringence, and classifies the exceptional traveling wave solutions.

Findings

Nonlinear solutions are exceptional traveling waves with constant speed.

No shock wave formation occurs in the studied configurations.

Vacuum birefringence is absent in Born-Infeld electrodynamics.

Abstract

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.