Numerical investigation of the Sine-Gordon equation with singularity solution behaviour at large t

The existance conditions of the periodic regimes inside the Josephson junction of finite length with microhomogeneity are studied. The sine-Gordon equation with singularity φtt=φxx-aφt-(1-μδ(x))sinφ+γ(t) is solved numerically. The solutions of kind ω+ρ(x,t) existance are established. Here ρ(x,t) is finite, periodic function of t. Similar solutions are observed also in homogeneous junction (when μ=0). We suceeded to excite the periodic regimes by varying the current function γ(t). New periodic regimes are obtained. The dependence ω on γ (current-voltage characteristics) graphes are presented. The approximate formulae ω(γ, α) of nonlinear part of the resistive branch is derived. The behaviour of ρ(x,t) for different branches of the current-voltage characteristics is studied