Stephanovich V A, Olchawa W, Kirichenko E V, Dugaev V K
Institute of Physics, University of Opole ul., Oleska 48, 45-052, Opole, Poland.
Department of Physics and Medical Engineering, Rzeszów University of Technology, al. Powstańców Warszawy 6, 35-959, Rzeszów, Poland.
Sci Rep. 2022 Sep 2;12(1):15031. doi: 10.1038/s41598-022-19332-z.
We examine the properties of a soliton solution of the fractional Schrö dinger equation with cubic-quintic nonlinearity. Using analytical (variational) and numerical arguments, we have shown that the substitution of the ordinary Laplacian in the Schrödinger equation by its fractional counterpart with Lévy index [Formula: see text] permits to stabilize the soliton texture in the wide range of its parameters (nonlinearity coefficients and [Formula: see text]) values. Our studies of [Formula: see text] dependence ([Formula: see text] is soliton frequency and N its norm) permit to acquire the regions of existence and stability of the fractional soliton solution. For that we use famous Vakhitov-Kolokolov (VK) criterion. The variational results are confirmed by numerical solution of a one-dimensional cubic-quintic nonlinear Schrödinger equation. Direct numerical simulations of the linear stability problem of soliton texture gives the same soliton stability boundary as within variational method. Thus we confirm that simple variational approach combined with VK criterion gives reliable information about soliton structure and stability in our model. Our results may be relevant to both optical solitons and Bose-Einstein condensates in cold atomic gases.
我们研究了具有立方 - 五次非线性的分数阶薛定谔方程孤子解的性质。通过解析（变分）和数值论证，我们表明，用具有 Lévy 指数[公式：见原文]的分数阶拉普拉斯算子替代薛定谔方程中的普通拉普拉斯算子，能够在其参数（非线性系数和[公式：见原文]）的广泛取值范围内稳定孤子结构。我们对[公式：见原文]依赖性（[公式：见原文]是孤子频率，N 是其范数）的研究，使得我们能够获取分数阶孤子解的存在区域和稳定性。为此，我们使用著名的瓦希托夫 - 科洛科洛夫（VK）准则。一维立方 - 五次非线性薛定谔方程的数值解证实了变分结果。孤子结构线性稳定性问题的直接数值模拟给出了与变分方法相同的孤子稳定性边界。因此，我们证实，简单的变分方法与 VK 准则相结合，能够为我们模型中的孤子结构和稳定性提供可靠信息。我们的结果可能与冷原子气体中的光学孤子和玻色 - 爱因斯坦凝聚体都相关。
