Solitary wave solutions and modulational instability analysis of the nonlinear Schrödinger equation with higher-order nonlinear terms in the left-handed nonlinear transmission lines

•Left-handed nonlinear transmission line is considered with higher order nonlinearity.•The nonlinear Schrödinger equation with cubic–quintic nonlinearities is introduced.•The generation of envelope solitons is studied through modulational instability. We report the modulational instability (MI) analysis for the modulation equations governing the propagation of modulated waves in a practical left-handed nonlinear transmission lines with series of nonlinear capacitance. Considering the voltage in the spectral domain and the Taylor series around a certain modulation frequency, we show in the continuum limit, that the dynamics of localized signals is described by a nonlinear Schrödinger equation with a cubic–quintic nonlinear terms. The MI process is then examined and we derive the gain spectra of MI for the generation of solitonlike-object in the transmission line metamaterials. We emphasize on the effect of losses on the MI gain spectra. An exact kink-darklike solutions is derived through the auxiliary equation method. It comes out that the width of the darklike solution decreases as the attenuation constant increases. Our theoretical solution is in good agreement with our numerical observation.