  We demonstrate that the plasmon in one-dimensional Coulomb interacting
electron fluids can develop a finite-momentum maxon-roton-like nonmonotonic
energy-momentum dispersion. Such an unusual nonmonotonicity arises from the
strongly interacting $1/r$ Coulomb potential going beyond the conventional band
linearization approximation used in the standard bosonization theories of
Luttinger liquids. We provide details for the nonmonotonic plasmon dispersion
using both bosonization and RPA theories. We also calculate the specific heat
including the nonmonotonicity and discuss possibilities for observing the
nonmonotonic plasmon dispersion in various physical systems including
semiconductor quantum wires, carbon nanotubes, and the twisted bilayer graphene
at sub-degree twist angles, which naturally realize one-dimensional domain-wall
states.
