When Does Linear Stability Not Exclude Nonlinear Instability ?

TL;DR

This paper reveals a mechanism where nonlinear coupling of internal modes can cause instability in stationary states that are linearly stable, demonstrated through three prototypical nonlinear Schrödinger equation examples.

Contribution

It introduces a nonlinear instability mechanism caused by internal mode coupling, expanding understanding beyond linear stability analysis in nonlinear Schrödinger equations.

Findings

Internal modes with negative energy lead to nonlinear instability.

Linear stability does not guarantee nonlinear stability.

Observed weak nonlinear instability in all three case studies.

Abstract

We describe a mechanism that results in the nonlinear instability of stationary states even in the case where the stationary states are linearly stable. This instability is due to the nonlinearity-induced coupling of the linearization's internal modes of negative energy with the wave continuum. In a broad class of nonlinear Schr{\"o}dinger (NLS) equations considered, the presence of such internal modes guarantees the nonlinear instability of the stationary states in the evolution dynamics. To corroborate this idea, we explore three prototypical case examples: (a) an anti-symmetric soliton in a double-well potential, (b) a twisted localized mode in a one-dimensional lattice with cubic nonlinearity, and (c) a discrete vortex in a two-dimensional saturable lattice. In all cases, we observe a weak nonlinear instability, despite the linear stability of the respective states.