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.
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