  We set up a hydrodynamic description of the non-equilibrium dynamics of
sine-Gordon quantum field theory for generic coupling. It is built upon an
explicit form of the Bethe Ansatz description of general thermodynamic states,
with the structure of the resulting coupled integral equations encoded in terms
of graphical diagrams. The resulting framework is applied to derive results for
the Drude weights of charge and energy. Quantities associated with the charge
universally have fractal dependence on the coupling, which is notably absent
from those associated with energy, a feature explained by the different roles
played by reflective scattering in transporting these quantities. We then
present far-from-equilibrium results, including explicit time evolution
starting from bipartite initial states and dynamical correlation functions. Our
framework can be applied to explore numerous other aspects of non-equilibrium
dynamics, which opens the way to a wide array of theoretical studies and
potential novel experimental predictions.
