We study the coupling between backward- and forward-propagating wave modes,with the same group velocity, in a composite right/left-handed nonlineartransmission line. Using an asymptotic multiscale expansion technique, wederive a system of two coupled nonlinear Schr{\"o}dinger equations governingthe evolution of the envelopes of these modes. We show that this systemsupports a variety of backward- and forward propagating vector solitons, of thebright-bright, bright-dark and dark-bright type. Performing systematicnumerical simulations in the framework of the original lattice that models thetransmission line, we study the propagation properties of the derived vectorsoliton solutions. We show that all types of the predicted solitons exist, butdiffer on their robustness: only bright-bright solitons propagate undistortedfor long times, while the other types are less robust, featuring shorterlifetimes. In all cases, our analytical predictions are in a very goodagreement with the results of the simulations, at least up to times of theorder of the solitons' lifetimes.
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