  We study inhomogeneous 1+1-dimensional quantum many-body systems described by
Tomonaga-Luttinger-liquid theory with general propagation velocity and
Luttinger parameter varying smoothly in space, equivalent to an inhomogeneous
compactification radius for free boson conformal field theory. This model
appears prominently in low-energy descriptions, including for trapped ultracold
atoms, while here we present an application to quantum Hall edges with
inhomogeneous interactions. The dynamics is shown to be governed by a pair of
coupled continuity equations identical to inhomogeneous Dirac-Bogoliubov-de
Gennes equations with a local gap and solved by analytical means. We obtain
their exact Green's functions and scattering matrix using a Magnus expansion,
which generalize previous results for conformal interfaces and quantum wires
coupled to leads. Our results explicitly describe the late-time evolution
following quantum quenches, including inhomogeneous interaction quenches, and
Andreev reflections between coupled quantum Hall edges, revealing a remarkably
universal dependence on details at stationarity or at late times out of
equilibrium.
