Traveling Waves in the Euler-Heisenberg Electrodynamics

We examine the possibility of travelling wave solutions within the nonlinear Euler-Heisenberg electrodynamics. Since this theory resembles in its form the electrodynamics in matter, it is a priori not clear if there exist travelling wave solutions with a new dispersion relation for $\omega(k)$ or if the Euler-Heisenberg theory stringently imposes $\omega=k$ for any arbitrary ansatz $\mathbf{E}(\xi)$ and $\mathbf{B}(\xi)$ with $\xi \equiv \mathbf{k}\cdot\mathbf{r} -\omega t$. We show that the latter scheme applies for the Euler-Heisenberg theory, but point out the possibility of new solutions with $\omega \neq k$ if we go beyond the Euler-Heisenberg theory, allowing strong fields. In case of the Euler-Heisenberg theory the quantum mechanical effect of the travelling wave solutions remains in $\hbar$ corrections to the energy density and the Poynting vector.