Self-trapped modes suffer critical collapse in two-dimensional cubic systems. To overcome such a collapse, linear periodic potentials or competing nonlinearities between self-focusing cubic and self-defocusing quintic nonlinear terms are often introduced. Here, we combine both schemes in the context of an unconventional and nonlinear fractional Schrödinger equation with attractive-repulsive cubic–quintic nonlinearity and an optical lattice. We report theoretical results for various two-dimensional trapped solitons, including fundamental gap and vortical solitons as well as the gap-type soliton clusters. The latter soliton family resembles the recently-found gap waves. We uncover that, unlike the conventional case, the fractional model exhibiting fractional diffraction order strongly influences the formation of higher band gaps. Hence, a new route for the study of self-trapped modes in these newly emergent higher band gaps is suggested. Regimes of stability and instability of all the soliton families are obtained with the help of linear-stability analysis and direct simulations.

自陷模式在二维立方系统中遭受严重崩溃。为了克服这种崩溃，经常引入自聚焦立方和自散焦五次非线性项之间的线性周期势或竞争性非线性。在这里，我们在非常规的非线性分数薛定ding方程，有吸引力的排斥立方-奎尼特非线性和光学晶格的背景下结合了这两种方案。我们报告了各种二维陷井孤子的理论结果，包括基本间隙和涡旋孤子以及间隙型孤子团簇。后者的孤子家族类似于最近发现的间隙波。我们发现，与常规情况不同，呈现分数衍射级数的分数模型强烈影响较高带隙的形成。因此，在这些新出现的较高带隙中，提出了一种研究自陷模式的新途径。借助于线性稳定性分析和直接仿真，可以获得所有孤子族的稳定性和不稳定性。