M. Lukas, D. Pelinovsky and P.G. Kevrekidis

Lyapunov-Schmidt reduction algorithm for three-dimensional discrete vortices

Physica D 237, 339-350 (2008)

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.

discrete nonlinear Schrodinger equation, discrete vortices, existence and stability, Lyapunov-Schmidt reductions, Mathematica symbolic computations