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We study the existence and stability of localized states in the discretenonlinear Schr{\"o}dinger equation (DNLS) on two-dimensional non-squarelattices. The model includes both the nearest-neighbor and long-rangeinteractions. For the fundamental strongly localized soliton, the resultsdepend on the coordination number, i.e., on the particular type of the lattice.The long-range interactions additionally destabilize the discrete soliton, ormake it more stable, if the sign of the interaction is, respectively, the sameas or opposite to the sign of the short-range interaction. We also explore morecomplicated solutions, such as twisted localized modes (TLM's) and solutionscarrying multiple topological charge (vortices) that are specific to thetriangular and honeycomb lattices. In the cases when such vortices areunstable, direct simulations demonstrate that they turn into zero-vorticityfundamental solitons.Comment: 17 pages, 13 figures, Phys. Rev.

Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity