Multiple-scale analysis of the parametric-driven sine-Gordon equation with phase shifts

In this article, we model the current and voltage across the weak link between two superconductors. This gives us a nonhomogeneous, nonlinear parametric-driven sine-Gordon equation with phase shifts. This model equation cannot be solved directly but can be approximated. For the approximations, we use two methods, and analytic perturbation method and the numerical approximation method known as the Runge-Kutta method. For the analytic method, we construct a perturbation expansion method with multiple-scale expansion. We discuss the parametric-driven in the sine-Gordon equation with phase shifts for the 0-π-0 junction. Further, we also describe the breathing modes for various order of perturbation. At the end, we compare the solutions obtained via perturbation and numerical methods of parametric-driven sine-Gordon equation with phase shifts. Finally, we concluded that the modes of the breathing decay to a constant in both cases. Also we found a good agreement between both approximate methods.