Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity

It is well known that the two-dimensional (2D) nonlinear Schrödinger equation (NLSE) with the cubic-quintic (CQ) nonlinearity supports a family of stable fundamental solitons, as well as solitary vortices (alias vortex rings), which are stable for sufficiently large values of the norm. We study stationary localized modes in a symmetric linearly coupled system of two such equations, focusing on asymmetric states. The model may describe “optical bullets” in dual-core nonlinear optical waveguides (including spatiotemporal vortices that have not been discussed before), or a Bose–Einstein condensate (BEC) loaded into a “dual-pancake” trap. Each family of solutions in the single-component model has two different counterparts in the coupled system, one symmetric and one asymmetric. Similarly to the earlier studied coupled 1D system with the CQ nonlinearity, the present model features bifurcation loops, for fundamental and vortex solitons alike: with the increase of the total energy (norm), the symmetric solitons become unstable at a point of the direct bifurcation, which is followed, at larger values of the energy, by the reverse bifurcation   restabilizing the symmetric solitons. However, on the contrary to the 1D system, both the direct and reverse bifurcation may be of the subcritical type, at sufficiently small values of the coupling constant, λλ. Thus, the system demonstrates a double bistability  for the fundamental solitons. The stability of the solitons is investigated via the computation of instability growth rates for small perturbations. Vortex rings, which we study for two values of the “spin”, s=1s=1 and 22, may be subject to the azimuthal instability, like in the single-component model. In particular, complete destabilization of asymmetric vortices is demonstrated for a sufficiently strong linear coupling. With the decrease of λλ, a region of stable asymmetric vortices appears, and a single region of bistability for the vortices is found. We also develop a quasi-analytical approach to the description of the bifurcations diagrams, based on the variational approximation. Splitting of asymmetric vortices, induced by the azimuthal instability, is studied by means of direct simulations. Interactions between initially quiescent solitons of different types are studied too. In particular, we confirm the prediction of the reversal of the sign of the interaction (attractive/repulsive for in-phase/out-of-phase pairs) for the solitons with the odd spin, s=1s=1, in comparison with the even values, s=0s=0and 2.

Research highlights
► Construction of symmetric and asymmetric fundamental and vortical 2-D solitons.
► Analytical approximations for the solitons agree with the numerical findings.
► Closed bifurcation loops are produced for solitons of all types.
► Azimuthally unstable asymmetric solitons split into sets of fragments.
► Study of Interactions between the solitons of various types is reported.