Dynamics of perturbations around inhomogeneous backgrounds in the HMF model

We investigate the dynamics of perturbations around inhomogeneous stationary states of the Vlasov equation corresponding to the Hamiltonian mean-field model. The inhomogeneous background induces a separatrix in the one-particle Hamiltonian system, and branch cuts generically appear in the analytic continuation of the dispersion relation in the complex frequency plane. We test the theory by direct comparisons with N-body simulations, using two families of distributions: inhomogeneous water-bags, and inhomogeneous thermal equilibria. In the water-bag case, which is not generic, no branch cut appears in the dispersion relation, whereas in the thermal equilibrium case, when looking for the root of the dispersion relation closest to the real axis, we have to consider several Riemann sheets. In both cases, we show that the roots of the continued dispersion relation give information that is useful for understanding the dynamics of a perturbation, although it is not complete.