On instability of excited states of the nonlinear Schrödinger equation

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.

Cited by in corpus (5)

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• On the Darboux and Birkhoff steps in the asymptotic stability of solitons
• On asymptotic stability of ground states of NLS with a finite bands periodic potential in 1D
• An invariant set in energy space for supercritical NLS in 1D
• Orbitally but not asymptotically stable ground states for the discrete NLS