content_image/356207.jpg content_image/356216.jpg content_image/356215.jpg
<img src='content_image/356207.jpg'>

Under these conditions the Abrikosov vortex contribution ϕ$_{V}$ ( ⃗ ρ ) to the total Josephson phase difference ϕ ( ⃗ ρ, t ) randomly depends on the position ⃗ ρ in the plane of the junction. 

The particular form of a single Abrikosov vortex contribution has been obtained previously in Refs. [19, 20]. 

The dynamics of a Josephson junction with randomly distributed Abrikosov vortices is described by an inhomogeneous sine-Gordon equation [1, 28, 29]: 

where 

<img src='content_image/356216.jpg'>

<img src='content_image/356215.jpg'>

Here V is the external dc voltage, c$_{0}$ is the velocity of Swihart waves, B$_{‖}$ is an external magnetic field applied parallel to the y -axis, λ$_{L}$$_{1}$ and λ$_{L}$$_{2}$ are the London penetration depths of the superconductors forming the junction, the parameter γ determines the decay of the electromagnetic waves in the junction and depends on the quasiparticle resistance of the junction in a subgap region, Φ$_{0}$ is the flux quantum . 

