On instability of excited states of the nonlinear Schr\"odinger equation

TL;DR

This paper introduces a new, stricter definition of linear stability for standing waves in the nonlinear Schrödinger equation and demonstrates that excited states are unstable under this criterion, suggesting it may be necessary for orbital stability.

Contribution

The paper proposes a novel, more comprehensive notion of linear stability for NLS standing waves and proves excited states are unstable under this new definition.

Findings

Excited states of NLS are not stable under the new stability notion.

The new stability criterion involves spectrum, kernel degeneracy, and eigenvalue signatures.

This stability notion is likely necessary for orbital stability.

Abstract

We introduce a new notion of linear stability for standing waves of the nonlinear Schr\"odinger 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.