W S$_{2}$ S$_{1}$ I  $_{L2}$  $_{L1}$ B$_{per}$ B$_{par}$ ! FIG. 1: Schematic cross-section of a Josephson tunnel junction with a misaligned Abrikosov vortex. δ is the misalignment length. λ$_{L}$$_{1}$ and λ$_{L}$$_{2}$ are the London penetration depths of the superconduc- tors forming the junction. The orientation of the two magnetic field, B$_{⊥}$ and B$_{‖}$ , is provided. 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 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 . 5 k$_{0}$ = 2 πB$_{‖}$ ( λ$_{L}$$_{1}$ + λ$_{L}$$_{2}$ ) Φ$_{0}$ , ω$_{0}$ = 2 eV ℏ . ∂ $^{2}$ϕ$_{1}$ ( ⃗ ρ, t ) ∂⃗ ρ 2 − 1 c 2 0 ∂ $^{2}$ϕ$_{1}$ ( ⃗ ρ, t ) ∂t 2 − γ c 2 0 ∂ϕ$_{1}$ ( ⃗ ρ, t ) ∂t = λ − 2 J sin[ ϕ$_{V}$ ( ⃗ ρ ) + k$_{0}$x + ω$_{0}$t + ϕ$_{1}$ ( ⃗ ρ, t )] , (2)