From Solitons to Rogue Waves in Nonlinear Left-Handed Metamaterials

In the present work, we explore soliton and rogue-like wave solutions in the transmission line analogue of a nonlinear left-handed metamaterial. The nonlinearity is expressed through a voltagedependent and symmetric capacitance motivated by the recently developed ferroelectric barium strontium titanate (BST) thin film capacitor designs. We develop both the corresponding nonlinear dynamical lattice, as well as its reduction via a multiple scales expansion to a nonlinear Schr\"odinger (NLS) model for the envelope of a given carrier wave. The reduced model can feature either a focusing or a defocusing nonlinearity depending on the frequency (wavenumber) of the carrier. We then consider the robustness of different types of solitary waves of the reduced model within the original nonlinear left-handed medium. We find that both bright and dark solitons persist in a suitable parametric regime, where the reduction to the NLS is valid. Additionally, for suitable initial conditions, we observe a rogue wave type of behavior, that differs significantly from the classic Peregrine rogue wave evolution, including most notably the breakup of a single Peregrine-like pattern into solutions with multiple wave peaks. Finally, we touch upon the behavior of generalized members of the family of the Peregrine solitons, namely Akhmediev breathers and Kuznetsov-Ma solitons, and explore how these evolve in the left-handed transmission line.