We study existence, bifurcation and stability of two-dimensional optical solitons in the framework of fractional nonlinear Schrödinger equation, characterized by its Lévy index, with self-focusing and self-defocusing saturable nonlinearities. We demonstrate that the fractional diffraction system with different Lévy indexes, combined with saturable nonlinearity, supports two-dimensional symmetric, antisymmetric and asymmetric solitons, where the asymmetric solitons emerge by way of symmetry breaking bifurcation. Different scenarios of bifurcations emerge with the change of stability: the branches of asymmetric solitons split off the branches of unstable symmetric solitons with the increase of soliton power and form a supercritical type bifurcation for self-focusing saturable nonlinearity; the branches of asymmetric solitons bifurcates from the branches of unstable antisymmetric solitons for self-defocusing saturable nonlinearity, featuring a convex shape of the bifurcation loops: an antisymmetric soliton loses its stability via a supercritical bifurcation, which is followed by a reverse bifurcation that restores the stability of the symmetric soliton. Furthermore, we found a scheme of restoration or destruction the symmetry of the antisymmetric solitons by controlling the fractional diffraction in the case of self-defocusing saturable nonlinearity.

中文翻译：

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分数衍射支持的二维光学孤子的存在，对称性打破分岔和稳定性

我们在分数非线性Schrödinger方程的框架下研究二维光学孤子的存在，分支和稳定性，该方程以其Lévy指数为特征，具有自聚焦和自散焦的饱和非线性。我们证明了具有不同Lévy指数的分数衍射系统，加上可饱和的非线性，支持二维对称，反对称和不对称孤子，其中非对称孤子通过对称性打破分叉而出现。随着稳定性的变化，出现了分叉的不同情况：随着孤子功率的增加，非对称孤子的分支从不稳定对称孤子的分支中分离出来，形成了超临界型分叉，用于自聚焦饱和非线性。非对称孤子的分支从不稳定的反对称孤子的分支中分叉，以实现自散焦的可饱和非线性，其特征是分叉环呈凸形：反对称孤子通过超临界分叉失去稳定性，然后进行反向分叉，从而恢复了对称孤子的稳定性。此外，我们发现了一种在自散焦可饱和非线性情况下通过控制分数衍射来恢复或破坏反对称孤子对称性的方案。随后是反向分叉，可恢复对称孤子的稳定性。此外，我们发现了一种在自散焦可饱和非线性情况下通过控制分数衍射来恢复或破坏反对称孤子对称性的方案。随后是反向分叉，可恢复对称孤子的稳定性。此外，我们发现了一种在自散焦可饱和非线性情况下通过控制分数衍射来恢复或破坏反对称孤子对称性的方案。