Dynamic Peierls-Nabarro equations for elastically isotropic crystals

The dynamic generalization of the Peierls-Nabarro equation for dislocations cores in an isotropic elastic medium is derived for screw and edge dislocations of the ``glide'' and ``climb'' type, by means of Mura's eigenstrains method. These equations are of the integrodifferential type and feature a nonlocal kernel in space and time. The equation for the screw differs by an instantaneous term from a previous attempt by Eshelby. Those for both types of edges involve in addition an unusual convolution with the second spatial derivative of the displacement jump. As a check, it is shown that these equations correctly reduce, in the stationary limit and for all three types of dislocations, to Weertman's equations that extend the static Peierls-Nabarro model to finite constant velocities.