Abstract
We study the coupling between the vortex phase and the cross phase in optical field propagation. The cross phase alone can split a vortex and steer the motion of the resulting fragments. With the cross phase present, the split vortices undergo a reciprocating motion; their separation oscillates in a way that does not occur otherwise. At a specific cross-phase coefficient, however, the vortex avoids splitting and preserves its elliptic shape during propagation. We attribute this to the anisotropic diffraction introduced by the cross-phase. When nonlinear self-focusing is included, these elliptic vortices evolve into stable solitons. These findings may find applications in beam shaping and optical manipulation.