Localized modes in a variety of driven long Josephson junctions with phase shifts

Abstract

We study both analytically and numerically the localized modes in long Josephson junctions with phase shift formations, so-called $$0{-}\pi {-}0$$ and $$0{-}\kappa $$ junctions. The system is described by an inhomogeneous sine-Gordon equation with a variety of time-periodic drives. Perturbation technique, together with multiple-scale expansions, is applied to obtain the amplitude of oscillations. It is observed that the obtained amplitude equations decay with time due to radiative damping and emission of high harmonic radiations. It is also observed that the energy taken away from the internal mode by radiation waves can be balanced by applying either direct or parametric drives. The appropriate external drives are applied to re-balance the dissipative and radiative losses. We discuss in detail the excitation by direct and parametric drives with frequencies to be either in the vicinity or double the natural frequency of the system. It is noted that the presence of external applied drives stabilizes the nonlinear damping, producing stable breather modes in long Josephson junctions. It is also noted that in the presence of parametric drives, the amplitudes of the driving forces are much more sensitive than in the case of external ac drives; that is, in the case of parametric drives, a small change in the amplitudes of the driving forces can make a drastic change in the system behavior and the system becomes unstable as compared to the case of the direct ac driving. Furthermore, we noticed that, in the presence of external driving, the driving effect is stronger for the case of driving frequency nearly equal to the system frequency as compared to that of the driving frequency nearly equal to twice the frequency of the oscillatory mode.