We analytically investigate a nonlinear fractional-order sine-Gordon (sG) equation. The derivatives considered herein, are taken in Caputo’s sense. The Laplace transform together with the Adomian decomposition method (LADM) is applied to attain analytical approximation of the aforesaid equation. The sG equation having Caputo derivative is solved in the order of series solutions and the results are confirmed by considering two examples with appropriate initial conditions. The numerical simulations are accomplished to compare with the analytical approximations, where qualitatively better agreements are achieved.

In this paper, we study, both analytically and numerically, the localized modes in long Josephson junctions in the presence of a variety of parametric drives. The phase-shift applied acts as a double-well potential which is known as the junction. The system is described by an inhomogeneous sine-Gordon equation that depicts the dynamics of long Josephson junctions with phase-shift. Using asymptotic analysis together with multiple scale expansion, we obtain the oscillation amplitudes for the junction, which show the decay behaviour with time due to radiative damping and the emission of high harmonic radiations. We found that the energy taken away from the internal modes by radiative waves can be balanced by some external drives. We also discuss in detail the excitation by parametric drives with frequencies to be either in the vicinity or double the natural frequency of the system. It is shown that the presence of externally applied drives stabilizes the nonlinear damping, producing stable breather modes in the junction. It is observed that, in the presence of external drives, the driving effect is stronger for a case of driving frequency nearly equal to the system’s natural frequency as compared to that of the driving frequency nearly equal to twice of the natural frequency of the oscillatory mode. In the numerical section, we calculate the Josephson voltage for the sine-Gordon equation with external drives. We observe that, an infinitely long Josephson junction with double well potential cannot be switched to a resistive state by an external drive with a frequency close to and double the systems eigenfrequency, provided that the driven amplitude is small enough. It is due to the fact that the higher harmonics with frequencies in the phonon band are excited in the form of continuous wave radiation.

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