Multi-hump soliton and rogue waves of the coupled nonlinear Schrödinger equations in nonlinear left-handed transmission line

This study derives a coupled system of nonlinear Schrödinger equations within a nonlinear left-handed transmission lattice to investigate localized waves and multi-hump solitons. Through a similarity transformation, the system is reduced to the Manakov system, enabling the identification of bright solitons, multi-hump, and rogue wave solutions by appropriately selecting constant parameters. Numerical simulations highlight the impact of wavenumber on the shape and amplitude of the double-hump single soliton, demonstrating that higher wavenumbers significantly alter the soliton’s shape. Furthermore, the study explores rogue wave solutions with a complex constant parameter, revealing that changes in both the constant parameter and the wavenumber introduce novel characteristics. These variations lead to the transformation of a bright rogue wave into a breather-like wave featuring multiple peaks. Additionally, the Peregrine soliton emerges within the system when a rogue wave solution is applied. These findings align with numerical simulations, revealing hump solitons and rogue waves characterized by two holes and one crest, resembling type I and II rogue waves. Evidently, the result shows that the multi-hump soliton and rogue wave structures are controllable. This work holds potential applications in metamaterial waveguides and microwaves, offering insight into extreme wave phenomena.