Three-dimensional hybrid solitary waves: Transverse vortex solitons stabilized by longitudinal parametric solitary waves

We show that the parametric process in quadratic nonlinear media supports three-dimensional (3D) hybrid solitary wave solution in which a transverse vortex solitons embedded in an infinite plane-wave background is sustained by a longitudinal parametric solitary wave. The structure of the parametric solitary wave results from the interplay of the quadratic nonlinearity and the temporal walk off (i.e., the velocity mismatch) between the interacting waves. The 3D hybrid solitary wave proved to be robust with respect to modulational instability, a feature that contrasts with previous studies on quadratic vortex solitons that revealed them to be always modulationally unstable. We show that the mechanism of stabilization of the vortex background lies on the temporal walkoff between the interacting waves that is able to drift the modulational instability out of the temporally localized structure that constitutes the 3D hybrid solitary wave.