  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 integro-differential type and feature a
non-local 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.
