  In this paper, we present two results on global continuation of monotone
front-type solutions to elliptic PDEs posed on infinite cylinders. This is done
under quite general assumptions, and in particular applies even to fully
nonlinear equations as well as quasilinear problems with transmission boundary
conditions. Our approach is rooted in the analytic global bifurcation theory of
Dancer and Buffoni--Toland, but extending it to unbounded domains requires
contending with new potential limiting behavior relating to loss of
compactness. We obtain an exhaustive set of alternatives for the global
behavior of the solution curve that is sharp, with each possibility having a
direct analogue in the bifurcation theory of second-order ODEs.
  As a major application of the general theory, we construct global families of
internal hydrodynamic bores. These are traveling front solutions of the full
two-phase Euler equation in two dimensions. The fluids are confined to a
channel that is bounded above and below by rigid walls, with incompressible and
irrotational flow in each layer. Small-amplitude fronts for this system have
been obtained by several authors. We give the first large-amplitude result in
the form of continuous curves of elevation and depression bores. Following the
elevation curve to its extreme, we find waves whose interfaces either overturn
(develop a vertical tangent) or become exceptionally singular in that the flow
in both layers degenerates at a single point on the boundary. For the curve of
depression waves, we prove that either the interface overturns or it comes into
contact with the upper wall.
