Exact solutions of the sine-Gordon equation describing oscillations in a long (but finite) Josephson junction

Abstract

Readily evaluated exact solutions of the sine-Gordon equation are presented for nonlinear standing-wave oscillations on a fixed length of a lossless Josephson transmission line with open-circuit boundary conditions at the ends. Three distinct species of standing waves are described: (i) plasma oscillation, (ii) breather oscillation, and (iii) fluxon oscillation. Fluxon oscillations can absorb power from an external source of bias current; for this case the volt-ampere characteristics relating bias current to average junction have been computed.

Bistability in the sine-Gordon equation: The ideal switch

“…where cn(·, m) is the cosine-amplitude Jacobi elliptic function of modulus m. According to [11], the resulting solution u(x, t) is called plasma oscillation and we have from (15)…”

“…Under boundary condition (3), in order to describe the periodic asymptotic regime reached in numerical simulations, we follow [11] and seek a solution…”

“…The parameters β and A are determined as before by requiring synchronization (11) and input datum (12), namely…”

Solitons and Long Josephson Junctions

“…The salient features that emerge from such an analysis, applied to the system described by Eq. (18), are [29]: (a) Phase-locked steps similar to the one shown in Fig. 3 exist at the fundamental frequency (n = m = 1) and at all odd subharmonics (n = 1; m = 1, 3, 5, .…”

“…Viewing Fulton's careful illustrations also renders intuitive the idea that a breather may be considered to be an oscillatory bound state of a kink and an anti-kink, an idea that has significant consequences for sine-Gordon dynamics. In fact, thoughtful study of Fulton's figures was instrumental in deriving the corresponding exact analytic solutions of the pure sine-Gordon equation on the finite interval, for arbitrary oscillation amplitudes, by Costabile et al [18].…”