Modulational instability and gap solitons in a finite Josephson transmission line

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Abstract
We concentrate on the modulational instability that may develop and perturb the gap soliton or nonlinear standing wave that appears in a finite Josephson transmission line [Phys. Rev. B 41, 10387 (1990)] when it has switched to a transmitting state. We first calculate theoretically the spatial dependence of the gap-soliton envelope or nonlinear standing wave inside the system, in terms of Jacobi elliptic functions. Our results fit reasonably well the envelope measured via our computer experiments on the dynamics of the system. Approaching the standing-wave structure in terms of two counterpropagating waves, we show that the evolution of their slowly varying amplitudes can be modeled by two coupled nonlinear Schrödinger equations. Then we calculate the critical wave number and growth rate of the instability for the two counter waves, which is three times larger than the growth rate of either wave alone. The corresponding critical temporal frequency, at which the modulational instability may appear, is in good agreement with the value determined from our numerical experiments.