Heteroclinic orbits for travelling kinks in differential, difference, and nonlocal wave equations

Travelling kinks in many nonlinear systems are described by heteroclinic orbits connecting saddle equilibrium points. In many systems, the equilibria are of the saddle-center type, and the kinks have oscillatory tails at infinity. I will discuss the conditions when the saddle-center points may admit a countable set of heteroclinic orbits with zero projection to the center manifold. The discussion is based on a number of physically relevant examples of differential, difference, and nonlocal wave equations.