Nonlinear dynamics of vortices in easy flow channels along grain   boundaries in superconductors
  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, $n\hbar\omega/2e$. 
A. Gurevich