  Guo--Tice formally established in 2011 that the Rayleigh--Taylor instability
inevitably occurs within stratified compressible viscous fluids in a slab
domain $\mathbb{R}^2\times (h_-,h_+)$, irrespecive of the presence of
interfacial surface tension, where the instability solutions are non-periodic
with respect to both horizontal spacial variables $x_1$ and $x_2$, by applying
a so-called ''normal mode'' method and a modified variational method to the
linearized (motion) equations. It is a long-standing open problem, however,
whether Guo--Tice's conclusion can be rigorously verified by the (original)
nonlinear equations. This challenge arises due to the failure of constructing a
growing mode solution, which is non-periodic with respect to both horizontal
spacial variables, to the linearized equations defined on a slab domain. In the
present work, we circumvent the difficulty related to growing mode solutions by
developing an alternative approximate scheme. In essence, our approach hinges
on constructing the horizontally periodic growing mode solution of the
linearized equations to approximate the {\it nonlinear} Rayleigh--Taylor
instability solutions, which do not exhibit horizontal periodicity. Thanks to
this new approximate scheme, we can apply Guo--Hallstrom--Spirn's bootstrap
instability method to the nonlinear equations in Lagrangian coordinates, and
thus prove Guo--Tice's conclusion. In particular, our approximate method could
also be applied to other instability solutions characterized by non-periodic
motion in a slab domain, such as the Parker instability and thermal
instability.
