  The Kawahara equation is a weakly nonlinear long-wave model of dispersive
waves that emerges when leading order dispersive effects are in balance with
the next order correction. Traveling wave solutions of the Kawahara equation
satisfy a fourth-order ordinary differential equation in which the traveling
wave speed is a parameter. The fourth order equation has Hamiltonian structure
and admits a two-parameter family of single-phase periodic solutions with
varying speed and Hamiltonian. A set of jump conditions is derived for pairs of
periodic solutions with equal speed and Hamiltonian. These are necessary
conditions for the existence of traveling waves that asymptote to the periodic
orbits at $\pm \infty$. Bifurcation theory and parameter continuation are used
to construct multiple solution branches of the jump conditions. For each pair
of compatible periodic solutions, the heteroclinic orbit representing the
traveling wave is constructed from the intersection of stable and unstable
manifolds of the periodic orbits. Each branch terminates at an
equilibrium-to-periodic solution in which the equilibrium is the background for
a solitary wave that connects to the associated periodic solution.
