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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. Intriguingly, we analytically observe that the product can be either simultaneously positive and negative, or positive positive, or even negative negative in different frequency bands. Our numerical results demonstrate 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.PACS 05.45,Yv · 04.20.Jb · 42.65.Tg

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