  We present new models of non-linear electromagnetism which satisfy the
Noether-Gaillard-Zumino current conservation and are, therefore, self-dual. The
new models differ from the Born-Infeld-type models in that they deform the
Maxwell theory starting with terms like $\lambda (\partial F)^{4}$. We provide
a recursive algorithm to find all higher order terms in the action of the form
$\lambda^{n} \partial ^{4n} F^{2n+2} $, which are necessary for the U(1)
duality current conservation. We use one of these models to find a self-dual
completion of the $\lambda (\partial F)^{4}$ correction to the open string
action. We discuss the implication of these findings for the issue of UV
finiteness of ${\cal N}=8$ supergravity.
