title: When linear stability does not exclude nonlinear instability.
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 continuous spectrum. In a broad class of nonlinear Schrödinger 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 antisymmetric 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.
authors.affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA.|Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USA.
authors.forename: P G
authors.initials: PG
authors.lastname: Kevrekidis
authors.affiliation: Department of Mathematics, McMaster University, Hamilton, Ontario, Canada L8S 4K1.|Department of Applied Mathematics, Nizhny Novgorod State Technical University, 24 Minin Street, Nizhny Novgorod, 603950, Russia.
authors.forename: D E
authors.initials: DE
authors.lastname: Pelinovsky
authors.affiliation: Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87544, USA.
authors.forename: A
authors.initials: A
authors.lastname: Saxena