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.

References in corpus (5)

• Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations
• Selection of the ground state for nonlinear Schroedinger equations
• On asymptotic stability in energy space of ground states for Nonlinear Schrödinger equations
• Relaxation of Excited States in Nonlinear Schrödinger Equations
• Stable Directions for Excited States of Nonlinear Schrödinger Equations

Cited by in corpus (10)

• On asymptotic stability in energy space of ground states for Nonlinear Schrödinger equations
• On asymptotic stability of standing waves of discrete Schrödinger equation in $\mathbb{Z}$
• On orbital instability of spectrally stable vortices of the NLS in the plane
• On the Darboux and Birkhoff steps in the asymptotic stability of solitons
• On asymptotic stability of moving ground states of the nonlinear Schrodinger equation
• 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
• Stabilization of small solutions of discrete NLS with potential having two eigenvalues
• Orbitally but not asymptotically stable ground states for the discrete NLS
• On stability of standing waves of nonlinear Dirac equations