Global bifurcation for monotone fronts of elliptic equations

In this talk, I will discuss some new results on global continuation of monotone front-type solutions to elliptic PDE posed on infinite cylinders.  This is done under quite general assumptions, and in particular applies even to fully nonlinear equations and variational problems with transmission conditions.  Our approach is rooted in the analytic global bifurcation theory of Dancer and Buffoni--Toland, but extending it to unbounded requires contending with new limiting behavior relating to loss of compactness.     

As a major application of the general theory, we construct a global family of internal hydrodynamic bores.  These are traveling front solutions of the full two-fluid 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 a continuous curve that is parameterized by the ratio of the downstream layer heights.   Following it to the extreme, one finds waves that are arbitrarily close to having points of horizontal stagnation.

This joint work with Ming Chen (Pitt) and Miles H. Wheeler (Bath)