Coupled backward- and forward-propagating solitons in a composite right- and left-handed transmission line

We study the coupling between backward- and forward-propagating wave modes, with the same group velocity, in a composite right- and left-handed nonlinear transmission line. Using an asymptotic multiscale expansion technique, we derive a system of two coupled nonlinear Schrödinger equations governing the evolution of the envelopes of these modes. We show that this system supports a variety of backward- and forward-propagating vector solitons of the bright-bright, bright-dark, and dark-bright type. Performing systematic numerical simulations in the framework of the original lattice that models the transmission line, we study the propagation properties of the derived vector soliton solutions. We show that all types of the predicted solitons exist, but differ on their robustness: Only bright-bright solitons propagate undistorted for long times, while the other types are less robust, featuring shorter lifetimes. In all cases, our analytical predictions are in very good agreement with the results of the simulations, at least up to times of the order of the solitons' lifetimes.