Soliton clusters in three-dimensional media with competing cubic and quintic nonlinearities

We introduce a class of robust soliton clusters composed of N fundamental solitons in three-dimensional media combining the self-focusing cubic and self-defocusing quintic nonlinearities. The angular momentum is lent to the initial cluster through staircase or continuous ramp-like phase distribution. Formation of these clusters is predicted analytically, by calculating an effective interaction Hamiltonian H-int. If a minimum of H-int is found, direct three-dimensional simulations demonstrate that, when the initial pattern is close to the predicted equilibrium size, a very robust rotating cluster does indeed exist, featuring persistent oscillations around the equilibrium configuration (clusters composed of N = 4, 5, and 6 fundamental solitons are investigated in detail). If a strong random noise is added to the initial configuration, the cluster eventually develops instability, either splitting into several fundamental solitons or fusing into a nearly axisymmetric vortex torus. These outcomes match the stability or instability of the three-dimensional vortex solitons with the same energy and spin; in particular, the number of the fragments in the case of the break-up is different from the number of solitons in the original cluster, being instead determined by the dominant mode of the azimuthal instability of the corresponding vortex soliton. The initial form of the phase distribution is important too: under the action of the noise, the cluster with the built-in staircase-like phase profile features azimuthal instability, while the one with the continuous distribution fuses into a vortex torus.