# An iterative scheme for the generalized Peierls-Nabarro model based on the inverse Hilbert transform

## TL;DR

This paper introduces a semi-analytical iterative scheme leveraging the inverse Hilbert transform to efficiently solve the generalized Peierls-Nabarro model, enabling detailed dislocation core structure analysis and external stress incorporation.

## Contribution

A novel semi-analytical iterative method based on the inverse Hilbert transform for solving the generalized Peierls-Nabarro model is proposed, reducing complex equations to a local fixed point iteration.

## Key findings

- Validated with 1D Peierls-Nabarro model examples
- Applied to dislocation core structures in Aluminium
- Discussed external stress incorporation and method limitations

## Abstract

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.