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.