On instability of excited states of the nonlinear Schrodinger equation

We introduce a new notion of linear stability for standing waves of the nonlinear Schrodinger equation (NLS) which requires not only that the spectrum of the linearization be real, but also that the generalized kernel be not degenerate and that the signature of all the positive eigenvalues be positive. We prove that excited states of the NUS are not linearly stable in this more restrictive sense. We then give a partial proof that this more restrictive notion of linear stability is a necessary condition to have orbital stability. (C) 2008 Elsevier B.V. All rights reserved.