Symmetric and asymmetric solitons and vortices in linearly coupled two-dimensional waveguides with the cubic-quintic nonlinearity (Q633715)

The authors study the system of two-dimensional, linearly coupled, nonlinear Schrödinger equations with cubic and quintic nonlinearity. Scalar equations admit stable fundamental solitons and stable solitary vortices (vortex rings). The coupling between two equations, however, may lead to asymmetric states because of various symmetry-breaking bifurcations. It is shown that each family of solutions in the single-component model has two different counterparts in the coupled system, one symmetric and the other one asymmetric. Bifurcation loops are formed where the symmetric solitons become unstable at a point of direct bifurcation which is followed at larger values of the energy by the reverse bifurcation restabilizing the symmetric solitons. Contrary to the one-dimensional model, where similar bifurcation loops were discovered by \textit{Z. Birnbaum} and \textit{B. A. Malomed} [Physica D 237, No. 24, 3252--3262 (2008; Zbl 1153.78323)], both the direct and reverse bifurcations may be of the subcritical type at sufficiently small values of the coupling constant. Hence, the system demonstrates a double bistability for the fundamental solitons. Vortex rings (of charge one and two) are shown to be the subject to the azimuthal instability, like in the single-component model. Complete destabilization of asymmetric vortices is demonstrated for a sufficiently strong linear coupling. With the decrease in the values of the coupling constant, a region of stable asymmetric vortices appears and a single region of bistability for the vortices is found. Numerical results are complemented by the semi-analytical description based on the variational approximation.