Localized vortices with a semi-integer charge in nonlinear dynamical lattices.

The topological charge S of vortexlike configurations in two-dimensional (2D) dynamical lattices need not necessarily be integer, nor is it a dynamical invariant. Accordingly, we demonstrate that the discrete nonlinear Schrödinger (DNLS) equation in 2D has stationary solutions in the form of a vortex with S=1/2, which does not exist in the model's continuum counterpart. Analysis of the DNLS equation linearized about the vortex shows that it is stable except for, possibly, extremely weak instabilities (at the level of numerical precision). Direct simulations of the full DNLS model in 2D show that the S=1/2 vortex soliton is a stable oscillating solution. This behavior of classical dynamical lattices is in contrast with a recently reported result by Clay et al. [Phys. Rev. Lett. 86, 4085 (2001)], according to which fractional charges in quantum lattices are subject to dynamical rearrangement into integer charges. We also consider S=1 discrete vortices that may be built as a pair of S=1/2 ones. These are different from the cross-shaped S=1 vortices that were recently found in the same 2D model. The S=1 vortices found in this work have larger energy and a slightly smaller stability range. We also find an analog of the S=1/2 vortices in the 1D DNLS model, which also turns out to be a stable oscillating soliton, different from the twisted localized modes recently found in the 1D model.