Unstable dispersive Hamiltonian evolution equations, such as semi-linear Klein-Gordon and Schroedinger equations, exhibit ”soliton”-like solutions. Amongst those one singles out the ground state, which has the lowest energy of all solitons. When the solutions are restricted to have energies at most slightly above that of the ground state, one obtains a trichotomy in forward time for this regime of energies. Recently, we found that this idea could be used to characterize solutions in the N-body problem. For N > 2, the question becomes quite complicated, and I will give some partial results.​ This is a joint work with Y. Deng.

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