M. Lukas, D. Pelinovsky and P.G. Kevrekidis
Lyapunov-Schmidt reduction algorithm for three-dimensional
discrete vortices
Physica D 237, 339-350 (2008)
Abstract:
We address the persistence and stability of three-dimensional vortex
configurations in the discrete nonlinear Schrodinger (NLS)
equation and develop a symbolic package based on Wolfram's
MATHEMATICA for computations of the Lyapunov--Schmidt reduction
method. The Lyapunov--Schmidt reduction method is a theoretical tool
which enables us to study continuations and terminations of the
discrete vortices for small coupling between lattice nodes as well
as the spectral stability of the persistent vortex configurations.
The method was developed earlier in the context of the
two-dimensional NLS lattice and applied to the on-site and off-site
configurations (called the vortex cross and the vortex cell) by
using semi-analytical computations. The present
treatment develops a full symbolic computational package which takes
a desired vortex configuration, performs a required number of
Lyapunov--Schmidt reductions and outputs the predictions on whether
the configuration persists in the three-dimensional lattice, whether
it is stable or unstable, and what approximations of unstable
eigenvalues in the linearized stability problem are. We report three
applications of the algorithm to particularly important vortex
configurations, such as the simple cube, the double cross, and the
diamond. At the simple cube and double cross configurations, we
identify exactly one vortex solution, which is stable for small
coupling between lattice nodes. At the diamond configuration, we
find that all vortex solutions are linearly unstable.
Keywords:
discrete nonlinear Schrodinger equation, discrete vortices,
existence and stability, Lyapunov-Schmidt reductions,
Mathematica symbolic computations