Interactions of three-dimensional solitons in the cubic-quintic model

We report results of a systematic numerical analysis of interactions between three-dimensional (3D) fundamental solitons, performed in the framework of the nonlinear Schr\"{o}dinger equation (NLSE) with the cubic-quintic (CQ) nonlinearity, combining the self-focusing and defocusing terms. The 3D NLSE with the CQ terms may be realized in terms of spatiotemporal propagation of light in nonlinear optical media, and in Bose-Einstein condensates, provided that losses may be neglected. The first part of the work addresses interactions between identical fundamental solitons, with phase shift $$ between them, separated by a finite distance in the free space. The outcome strongly changes with the variation of $\varphi$: in-phase solitons with $\varphi =0$, or with sufficiently small $\varphi$, merge into a single fundamental soliton, with weak residual oscillations in it (in contrast to the merger into a strongly oscillating breather, which is exhibited by the 1D version of the same setting), while the choice of $$ leads to fast separation between mutually repelling solitons. At intermediate values of $\varphi$, such as $\varphi =\pi /2$, the interaction is repulsive too, breaking the symmetry between the initially identical fundamental solitons, there appearing two solitons with different total energies (norms). The symmetry-breaking effect is qualitatively explained, similar to how it was done previously for 1D solitons. In the second part of the work, a pair of fundamental solitons trapped in a 2D potential is considered. It is demonstrated that they may form a slowly rotating robust \textquotedblleft molecule", if aninitial kicks are applied to them in opposite directions, perpendicular to the line connecting their centers.