  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.
