Nondegenerate and almost hexagonal skyrmion lattices

We obtain the lowest energy solutions for the skymion field equations and their corresponding vortex structures. Two nondegenerate solutions emerge with their vortex swirls in opposite directions. The solutions are associated with an extremum property, which favors an array of almost hexagonal shape. We predict that a regular hexagonal lattice must have a mix of skyrmions of both swirls. Although our solutions could not keep the norm of the magnetization constant at unity, their greatest deviation from unity occurred in regions where the spins are far from planar; we show how to improve this situation.