Modulation effect on a perturbed sine-Gordon system with a long overlap Josephson junction

We report numerical simulations on the phase locking of solitons generated by rf excitations. In addition to applying a dc magnetic field and an rf signal, we also employed an ac frequency modulation field on a long overlap Josephson junction. The normalized frequency of the rf signal applied on the boundary of the junction is ω₁=0.214, while that for the low modulation field is ω₂=0.002. The analysis indicated that the rf signal applying on both ends of the junction will excite two breather modes, and the total energy per unit length of soliton is mostly embodied in the second harmonic waves of ω₁, based on the Fourier analysis. The effect of the low-frequency magnetic field will excite low-subband mixing waves with or without the presence of the dc magnetic field