分数阶非线性薛定谔方程中的涡旋孤子
Vortex solitons in fractional nonlinear Schrödinger equation with the cubic-quintic nonlinearity
摘要：
研究了一类分数阶非线性薛定谔方程（NLSE）的涡旋孤子解的存在性和稳定性，该方程具有竞争的五阶非线性项和Lévy指数（分数阶），其值为1 \leq α \leq 2.以数值形式构造了涡度s = 1，2和3的环形涡系族。与通常的二维NLSE（对应于α = 2）不同，在分数阶模型中，VS存在于总功率P的有限阈值之上。针对线性化方程所控制的小扰动，研究了VS解的稳定性，并通过直接模拟进行了证实。不稳定的VS被打破的方位角扰动成几个片段，其数量是由增长最快的本征模的小扰动。对于所有s = 1、2和3，用P定义的稳定区域随着α从1增加到2而扩大，除了s = 2时在1 \leq α \leq 1.3的区间内急剧收缩。
英文摘要：
We address the existence and stability of vortex-soliton (VS) solutions of the fractional nonlinear Schrödinger equation (NLSE) with competing cubic-quintic nonlinearities and the Lévy index (fractionality) taking values 1 \leqα\leq2. Families of ring-shaped VSs with vorticities s = 1,2, and 3 are constructed in a numerical form. Unlike the usual two-dimensional NLSE (which corresponds to α = 2), in the fractional model VSs exist above a finite threshold value of the total power,P. Stability of the VS solutions is investigated for small perturbations governed by the linearized equation, and corroborated by direct simulations. Unstable VSs are broken up by azimuthal perturbations into several fragments, whose number is determined by the fastest growing eigenmode of small perturbations. The stability region, defined in terms of P, expands with the increase of α from 1 up to 2 for all s = 1, 2, and 3, except for steep shrinkage for s = 2 in the interval of 1\leqα\leq1.3.
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