  For a natural number $m \ge 2$, we study $m$ layers of finite depth,
horizontally infinite, viscous, and incompressible fluid bounded below by a
flat rigid bottom. Adjacent layers meet at free interface regions, and the top
layer is bounded above by a free boundary as well. A uniform gravitational
field, normal to the rigid bottom, acts on the fluid. We assume that the fluid
mass densities are strictly decreasing from bottom to top and consider the
cases with and without surface tension acting on the free surfaces. In addition
to these gravity-capillary effects, we allow a force to act on the bulk and
external stress tensors to act on the free interface regions. Both of these
additional forces are posited to be in traveling wave form: time-independent
when viewed in a coordinate system moving at a constant, nontrivial velocity
parallel to the lower rigid boundary. Without surface tension in the case of
two dimensional fluids and with all positive surface tensions in the higher
dimensional cases, we prove that for each sufficiently small force and stress
tuple there exists a traveling wave solution. The existence of traveling wave
solutions to the one layer configuration ($m=1$) was recently established and,
to the best of our knowledge, this paper is the first construction of traveling
wave solutions to the incompressible Navier-Stokes equations in the $m$-layer
arrangement.
