Lyapunov–Schmidt reduction algorithm for three-dimensional discrete vortices
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abstract
We address persistence and stability of three-dimensional vortex
configurations in the discrete nonlinear Schr\"{o}dinger (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 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 waveform
at the anti-continuum limit of uncoupled sites, performs a required number of
Lyapunov--Schmidt reductions and outputs the predictions on whether the
configuration persists, for finite coupling, in the three-dimensional lattice
and whether it is stable or unstable. It also provides approximations for the
eigenvalues of the linearized stability problem. We report a number of
applications of the algorithm to important multi-site configurations, such as
the simple cube, the double cross, and the diamond. For each three-dimensional
configuration, we identify exactly one solution, which is stable for small
coupling between lattice nodes.
