D.E. Pelinovsky and J. Yang
On transverse stability of discrete line solitons
Physica D 255, 1-11 (2013)
Abstract:
We obtain sharp criteria for transverse stability and instability
of line solitons in the discrete nonlinear Schrodinger equations on
one- and two-dimensional lattices near the anti-continuum limit.
On a two-dimensional lattice, the fundamental line soliton is
proved to be transversely stable (unstable) when it bifurcates
from the X (G) point of the dispersion surface. On a one-dimensional
(stripe) lattice, the fundamental line soliton is proved to be
transversely unstable for both signs of transverse dispersion. If this
transverse dispersion has the opposite sign to the discrete
dispersion, the instability is caused by a resonance between isolated
eigenvalues of negative energy and the continuous spectrum of
positive energy. These results hold for both focusing and defocusing
nonlinearities via a staggering transformation. When the line
soliton is transversely unstable, asymptotic expressions for unstable
eigenvalues are also derived. These analytical results are compared
with numerical results, and perfect agreement is obtained.
Keywords:
discrete nonlinear Schrodinger equations, line solitons, transverse stability, negative index theory