I will talk about the recent development on the time-asymptotic stability of composite waves consisting a degenerate viscous Oleinik shock and a rarefaction for the non-convex scalar conservation laws. The main difficulty is due to the incompatibility of the stability proof framework of individual viscous shock and rarefaction wave. We develop a new type of a-contraction method and the choice of the weight function is quite different, but consistent with the classical one for the Burgers equation. The Oleinik shock wave strength can be arbitrarily large. Another difficulty comes from that the Oleinik shock and rarefaction waves are always attached for the non-convex scalar conservation laws such that the time-dependent shift function needs to be equipped to both Oleinik shock and rarefaction waves. Moreover, we need to treat the subtle wave interactions. Note that this is the first resolution concerning the asymptotic toward the superposition of a degenerate viscous shock and a rarefaction for the scalar viscous conservation laws.