  We analyze the component structure of models for 4D N = 1 supersymmetric
nonlinear electrodynamics that enjoy invariance under continuous duality
rotations. The N = 1 supersymmetric Born-Infeld action is a member of this
family. Such dynamical systems have a more complicated structure, especially in
the presence of supergravity, as compared with well-studied effective
supersymmetric theories containing at most two derivatives (including nonlinear
Kahler sigma-models). As a result, when deriving their canonically normalized
component actions, it becomes impractical and cumbersome to follow the
traditional approach of (i) reducing to components; and then (ii) applying a
field-dependent Weyl and local chiral transformation. It proves to be more
efficient to follow the Kugo-Uehara scheme which consists of (i) extending the
superfield theory to a super-Weyl invariant system; and then (ii) applying a
plain component reduction along with imposing a suitable super-Weyl gauge
condition. Here we implement this scheme to derive the bosonic action of
self-dual supersymmetric electrodynamics coupled to the dilaton-axion chiral
multiplet and a Kahler sigma-model. In the fermionic sector, the action
contains higher derivative terms. In the globally supersymmetric case, a
nonlinear field redefinition is explicitly constructed which eliminates all the
higher derivative terms and brings the fermionic action to a one-parameter
deformation of the Akulov-Volkov action for the Goldstino. The Akulov-Volkov
action emerges, in particular, in the case of the N = 1 supersymmetric
Born-Infeld action.
