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

TL;DR

This paper presents a systematic numerical study of three-dimensional soliton interactions in the cubic-quintic nonlinear Schrödinger equation, revealing how phase differences influence merging, separation, and symmetry-breaking, with potential applications in optics and Bose-Einstein condensates.

Contribution

It provides the first detailed analysis of 3D soliton interactions in the cubic-quintic model, including effects of phase shifts and formation of rotating soliton molecules.

Findings

In-phase solitons merge into a single soliton.

Out-of-phase solitons with phase difference π repel each other.

Intermediate phase differences cause symmetry breaking and energy asymmetry.

Abstract

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 $% \varphi $ 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 $% \varphi =\pi $ 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.