SG-kink dynamics under dissipation and external regular and random forces

Within the formalism of the Fokker-Planck equation, the influence of nonstationaryregular external force, random force, and dissipation on the kink dynamics is investigatedin the sine-Gordon model. The evolution equation of the kink momentum is written inthe form of the stochastic differential equation in the Stratonovich sense within theframework of the well-known McLaughlin and Scott energy approach. Thecorresponding Fokker-Planck equation for the momentum distribution function coincideswith the equation describing the Ornstein-Uhlenbek process with a regular nonstationaryexternal force. The influence of the nonlinear stochastic effects on the kink dynamics isdescribed in terms of the nonlinear Fokker-Planck equation with the drift coefficientdependent on the first moment of the distribution function. Exact solutions of both linearand nonlinear Fokker-Planck equations are presented. The average value and the varianceof the kink momentum are obtained in explicit form using the above exact solutions.Examples are considered which demonstrate the influence of the external regular andrandom forces on the evolution of the average value and variance of the kink momentum.