Investigation of tripled sine-Gordon equation: Localized modes in multi-stacked long Josephson junctions

This study systematically examines the behavior of multi-stacked long Josephson junctions (multi-sLJJs) under various magnetic inductances and a variety of drives. To investigate the localized modes in the multi-sLJJs, the tripled sine-Gordon equation (sGE) with a phase shift formation recognized as 0-π\pi -0 junction is analyzed. The amplitude equations are derived through asymptotic analysis. It is revealed that synchronized oscillations occur in the multi-sLJJs under both strong and weak couplings; however, the system experiences exponential decay in the absence of external drives. The stack naturally exhibits damping due to the energy produced by the oscillating mode through radiation, which arises from the higher harmonic excitation introduced into the system. It is observed that when an AC drive is present, the system exhibits synchronous oscillations for both strong and weak inductances but eventually becomes stable. However, for parametric drives under both coupling conditions, the system experiences exponential decay. Our findings suggest that weak coupling renders parametric drives ineffective. Nevertheless, numerical simulations demonstrate that a large driving amplitude induces oscillations in the system and causes it to approach a steady position under strong coupling.