Stability of bound states of Hamiltonian PDEs in the degenerate cases

We consider a Hamiltonian systems which is invariant under a one-parameter unitary group and give a criterion for the stability and instability of bound states for the degenerate case. We apply our theorem to the single power nonlinear Klein–Gordon equation and the double power nonlinear Schrödinger equation. © 2012 Elsevier Inc. All rights reserved.