Envelope solitons in a left-handed nonlinear transmission line with Josephson junction

We consider a nonlinear left-handed transmission line that incorporates an array of Josephson junctions in its periodic lattice structure. We show that the system dynamics is described by a discrete sine-Gordon-like equation, where the left-handedness of the lattice manifests in the form of a non-standard second-time- derivative term. Since this modified discrete sine-Gordon equation has not yet been extensively studied in the literature, this paper opens up the possibility of additional mathematical analysis. It is also intriguing that by means of a semi-discrete approximation we can derive a nonlinear Schrödinger equation and thus show that the system supports both bright and dark envelope soliton solutions depending on the choice of carrier frequency. The left-handedness of the network is explicitly confirmed in numerical simulations which demonstrate the backward propagation of the bright and dark soliton, in good agreement with analytical predictions.