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A necessary condition for the existence of a skyrmion in two-componentBose-Einstein condensates with $\mathrm{SU(2)}$ symmetry was recently providedby two of the authors [Phys. Rev. Lett. {\bf 97}, 080403 (2006)], by mappingthe problem to a classical particle in a potential subject to time-dependentdissipation. Here we further elaborate this approach. For two classes ofmodels, we demonstrate the existence of the critical dissipation strength abovewhich the skyrmion solution does not exist. Furthermore, we discuss the localstability of the skyrmion solution by considering the second-order variation. Asufficient condition for the local stability is given in terms of theground-state energy of a one-dimensional quantum-mechanical Hamiltonian. Thiscondition requires a minimum number of bosons, for a certain class of the trappotential. In the optimal case, the minimum number of bosons can be as small$\sim 10^4$.

Skyrmion in spinor condensates and its stability in trap potentials