  A new semi-analytical iterative scheme is proposed in this work for solving
the generalized Peierls-Nabarro model. The numerical method developed here
exploits certain basic properties of the Hilbert transform to achieve the
desired reduction of the non-local and non-linear equations characterizing the
generalized Peierls-Nabarro model to a local fixed point iteration scheme. The
method is validated with simple examples involving the 1D Peierls-Nabarro model
corresponding to a sinusoidal stacking fault energy, and with calculations of
the core structure of both edge and screw dislocations on the close-packed
$\{111\}$ planes in Aluminium. An approximate technique to incorporate external
stresses within the framework of the proposed iterative scheme is also
discussed with applications to the equilibration of a dislocation dipole.
Finally, the advantages, limitations and avenues for future extension of the
proposed method are discussed.
