Backward- and forward-wave soliton coexistence due to second-neighbor coupling in a left-handed transmission line

Abstract

We study analytically and numerically the impact of the second-neighbor interactions on the propagation of an initial wave-packet through a coupled nonlinear left-handed transmission line. We start with a detailed analysis of the dispersion curve and group velocity. We then use the quasi-discrete approximation with respect to long wavelength transverse perturbations to reduce the nonlinear discrete model of the lattice to a nonlinear Schrödinger equation. A detailed analysis of the product of the group velocity dispersion (P) and the nonlinear term (Q) is presented with respect to the relevant parameters of the system. Here we see the important role of the second-neighbor coupling, which makes the dispersion curve non-monotonic. Thus, for a given frequency, the group velocity can be both positive and negative, while for each choice the product, PQ, can be either positive or negative. Our numerical results confirm that for sufficiently strong second-neighbor coupling, bright backward-wave pulses and dark solitons can coexist at a common frequency, and these travel in opposite directions through the lattice.