Abstract This research is devoted to analyze a conformable fractional-order model of nonlinear two-dimensional transmission line metamaterials. The conformable fractional derivative is utilized to broaden classical differentiation while maintaining essential aspects, including the chain rule. Transmission line metamaterials, as intentionally designed structures, offer a robust framework for examining wave transmission in nonlinear dispersive media. The suggested electrical model facilitates understanding the behavior of creation and transmission of solitons, examined via voltage wave dynamics in circuit-based representations including inductors and capacitors. Due to the electromagnetic characteristics, metamaterials possess considerable promise for use in microwave engineering, signal processing, and communications. Three analytical techniques are utilized to investigate the model’s dynamics: the sine–Gordon expansion method, the fractional sub-equation approach, and the tanh method. Each approach was implemented within the fractional-order interval of the derivative parameter, 0 < α ≤ 1. The derived solutions demonstrated both singular and multiple soliton configurations, essential for the development of sophisticated waveguides and the improvement of signal transmission in telecommunications. The comparison research revealed that all three strategies effectively generated soliton solutions; however, the sine–Gordon expansion method was more beneficial, as it produced more generalized solution forms. This underscores its adaptability and wider applicability in the modeling and construction of fractional-order metamaterial systems.
Almetwally et al. (2026) studied this question.