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

TL;DR

This paper investigates the existence, stability, and bifurcation behavior of symmetric and asymmetric solitons and vortices in a coupled 2D nonlinear Schrödinger system with cubic-quintic nonlinearity, relevant to optical and BEC systems.

Contribution

It introduces a detailed analysis of bifurcation loops and bistability phenomena in coupled 2D CQ-NLS models, including vortex dynamics and a variational approximation approach.

Findings

Identification of bifurcation loops for solitons and vortices.

Observation of double bistability in fundamental solitons.

Analysis of vortex splitting due to azimuthal instability.

Abstract

It is well known that the two-dimensional (2D) nonlinear Schr\"odinger 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 were not 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, the system may demonstrate 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 = 1 and 2, may be subject to the azimuthal instability, like in the single-component model. 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.