  We consider the existence and spectral stability of periodic multi-pulse
solutions in Hamiltonian systems which are translation invariant and
reversible, for which the fifth-order Korteweg-de Vries equation is a
prototypical example. We use Lin's method to construct multi-pulses on a
periodic domain, and in particular demonstrate a pitchfork bifurcation
structure for periodic double pulses. We also use Lin's method to reduce the
spectral problem for periodic multi-pulses to computing the determinant of a
block matrix, which encodes both eigenvalues resulting from interactions
between neighboring pulses and eigenvalues associated with the essential
spectrum. We then use this matrix to compute the spectrum associated with
periodic single and double pulses. Most notably, we prove that brief
instability bubbles form when eigenvalues collide on the imaginary axis as the
periodic domain size is altered. These analytical results are all in good
agreement with numerical computations, and numerical timestepping experiments
demonstrate that these instability bubbles correspond to oscillatory
instabilities.

Dynamical Systems