Analytical solution of fluxons in a non-homogeneous Josephson junction

In order to study the fluxon behavior in a Josephson junction, a dielectric with variable thickness was inserted between two superconductors, which can be position-dependent. An analytical solution of the sine-Gordon equation is obtained. Subsequently, the relation between the kinetic and potential energy of a fluxon is obtained along with the analytic relationships between the velocity of the fluxon and the dielectric thickness. Finally, an exact topological soliton solution is obtained for the sine-Gordon equation with and without strong perturbations through the use of the ansatz method.