We introduce a new notion of linear stability for standing waves of the nonlinear Schrödinger 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 NLS 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.
On instability of excited states of the nonlinear Schrödinger equation / Cuccagna, Scipio. - In: PHYSICA D-NONLINEAR PHENOMENA. - ISSN 0167-2789. - STAMPA. - 238:1(2009), pp. 38-54. [10.1016/j.physd.2008.08.010]
On instability of excited states of the nonlinear Schrödinger equation
CUCCAGNA, Scipio
2009
Abstract
We introduce a new notion of linear stability for standing waves of the nonlinear Schrödinger 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 NLS 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.
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