Patterns of rational solutions in a split-ring-resonator-based left-handed coplanar waveguide

The exploration of rational solutions of first and second orders, along with the investigation of modulation instability, has been conducted in the left-handed coplanar waveguide based on split-ring resonators. This study is inspired by the research of Abbagari et al. (0000), where solitonic rogue wave structures were derived as manifestations of the growth rate of modulation instability. Under this argument, we have used the perturbations method to derive the Kundu–Eckhaus equation to analyze the characteristics of the high-order rogue waves. Beside these findings, we have realized that rogue wave structures are propagated in the left-handed frequency bands. We also notice that modulation instability growth develops in the frequency bands when the product of the nonlinearity coefficient and dispersion coefficient is positive. Through a numerical simulation, we have developed the rogue wave objects to confirm our analytical predictions. Another significant aspect addressed in this study is the sensitivity of both modulation instability and higher-order rogue waves to the normalized parameter introduced through the third-order expansion of the voltage-dependent capacitance and perturbed wave number. The long-lived results have been equally validated for specific times of propagation. These results could be used in the future in left-handed metamaterials for several applications.