Breather and solitonic behavior of parametric Sine–Gordon equation with phase-shift and driven term

This study presents a comprehensive analytical and computational investigation of the nonlinear parametric sine-Gordon equation (sGE) with a driven term and phase shift. The sGE model captures the current and voltage dynamics across a weak connection between two superconductors, providing valuable insights into the behavior of Josephson Junction systems. To validate the results, two primary methodologies are employed for approximating solutions to the sGE. Analytically, a perturbative expansion combined with multiple-scale analysis is developed to derive system dynamics up to the fifth-order expansion. Numerically, the explicit finite difference scheme and the fourth-order Runge–Kutta finite difference method are implemented. The model equation is formulated for a 0−π−0 junction with appropriate initial and boundary conditions. Furthermore, the stability of the numerical scheme is rigorously analyzed, accounting for constraints imposed by the nonlinear terms. The findings reveal that the breathing modes of oscillation decay towards a constant state, with strong agreement observed between the analytical and numerical results. Additionally, one- and two-dimensional numerical computations are performed to enhance the clarity and depth of the analysis. This research significantly contributes to the fields of superconductivity and nonlinear dynamics, offering novel insights into the complex behavior of coupled systems and advancing the understanding of Josephson Junction dynamics.