  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.
