Instabilities of breathers in a finite NLS lattice

We study some aspects of the dynamics of unstable breathers in a three-site discrete cubic NLS chain with Dirichlet boundary conditions. We view breathers as fixed points of the energy in the reduced phase space obtained by eliminating directions related to the global phase symmetry of the system. We use a combination of numerical calculations and Morse-theoretical arguments to see that there are two breathers that correspond to critical energies where the energy hypersurface changes its connectivity. These breathers are elliptic–hyperbolic fixed points of the reduced four-dimensional system. We compute the periodic orbits in their center manifolds (Lyapunov orbits) and see evidence for homoclinic intersections of their stable and unstable manifolds. We also examine the possibility of heteroclinic connections between Lyapunov orbits, these however appear not to exist for the energies near the energy where the energy hypersurface becomes connected.