Dynamics of coupled sine-Gordon equations: Inductively stacked long Josephson junctions with heterogeneous drives

The article presents the systematic study of the stacked long Josephson junctions in presence of different magnetic inductances. The coupled sine-Gordon equation along with a phase-shift formation is considered to study localized modes in stacks. The asymptotic analysis is applied to investigate analytically the nonlinear amplitude equations by using the spatially continuous and discrete spectrums. It is observed that in the absence of external drives, the system decays exponentially in the case of both strong and weak magnetic inductance, where the synchronized oscillations are obtained. However, in the presence of AC-drive, the system oscillates synchronically and goes to a stable state after a long time. It is revealed that, when the parametric drives are applied, the system decays exponentially in the existence of small driving amplitude. However, when the driving amplitude is large enough, the system oscillates for an infinitely long time.