P.G. Kevrekidis, D.E. Pelinovsky, and A. Saxena
When linear stability does not exclude nonlinear instability
Physical Review Letters 114, 214101 (2015) (6 pages)
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 Schrodinger (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.
nonlinear Schrodinger equation, instability, negative Krein signature, internal modes.