  A theory of nonlinear dynamics of mixed Abrikosov vortices with Josephson
cores (AJ vortices) on low-angle grain boundaries (GB) in superconductors is
proposed. Dynamics and pinning of AJ vortices determine the in-field current
transport through GB and the microwave response of polycrystal in the crucial
misorientation range $\vartheta < 20-30^{\circ}$ of the exponential drop of the
local critical current density $J_b(\vartheta)$ through GB. An exact solution
for an overdamped periodic AJ vortex structure driven along GB by an arbitrary
time dependent transport current in a dc magnetic field $H>H_{c1}$ is obtained.
The dynamics of the AJ vortex chain is parameterized by solutions of two
coupled first order nonlinear differential equations which describe
self-consistently the time dependence of the vortex velocity and the AJ core
length. Exact formulas for the dc flux flow resistivity $R_f(H)$, and the
nonlinear voltage-current characteristics are obtained. Dynamics of the AJ
vortex chain driven by superimposed ac and dc currents is considered, and
general expressions for a linear complex resistivity $R(\omega)$ and
dissipation of the ac field are obtained. A flux flow resonance is shown to
occur at large dc vortex velocities $v$ for which the imaginary part of
$R(\omega)$ has peaks at the "washboard" ac frequency $\omega_0=2\pi v/a$,
where $a$ is the inter vortex spacing. This resonance can cause peaks and
portions with negative differential conductivity on the averaged dc
voltage-current (V-I) characteristics. Ac currents of large amplitude cause
generation of higher voltage harmonics and phase locking effects which manifest
themselves in steps on the averaged dc I-V curves at the Josephson voltages,
$n\hbar\omega/2e$.
