  We prove the time-asymptotic stability of composite waves consisting of the
superposition of a viscous shock and a rarefaction for the one-dimensional
compressible barotropic Navier-Stokes equations. Our result solves a
long-standing problem first mentioned in 1986 by Matsumura and Nishihara in
[25]. The same authors introduced it officially as an open problem in 1992 in
[26] and it was again described as very challenging open problem in 2018 in the
survey paper [23]. The main difficulty is due to the incompatibility of the
standard anti-derivative method, used to study the stability of viscous shocks,
and the energy method used for the stability of rarefactions. Instead of the
anti-derivative method, our proof uses the $a$-contraction with shifts theory
recently developed by two of the authors. This method is energy based, and can
seamlessly handle the superposition of waves of different kinds.
