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We consider a prototypical dynamical lattice model, namely, the discretenonlinear Schroedinger equation on nonsquare lattice geometries. We present asystematic classification of the solutions that arise in principalsix-lattice-site and three-lattice-site contours in the form of both discretemultipole solitons and discrete vortices. Additionally to identifying thepossible states, we analytically track their linear stability bothqualitatively and quantitatively. We find that among the six-siteconfigurations, the hexapole of alternating phases, as well as the vortex oftopological charge S=2 have intervals of stability; among three-site states,only the vortex of topological charge S=1 may be stable in the case of focusingnonlinearity. These conclusions are confirmed both for hexagonal and forhoneycomb lattices by means of detailed numerical bifurcation analysis of thestationary states from the anticontinuum limit, and by direct simulations tomonitor the dynamical instabilities, when the latter arise. The dynamics reveala wealth of nonlinear behavior resulting not only in single-site solitary waveforms, but also in robust multisite breathing structures.

Discrete solitons and vortices in hexagonal and honeycomb lattices: Existence, stability, and dynamics