Title: Suppression of soliton collapses, modulational instability, and rogue-wave excitation in two-Lévy-index fractional Kerr media Show Chinese title

Title: 两 Levy 指数分数 Kerr 媒质中孤子塌缩、调制不稳定性及 rogue 波激发的抑制

Abstract: s in laser systems with two fractional-dispersion/diffraction terms, quantified by their L\'{e}vy indices, $\alpha_{1}\, \alpha_{2}\in (1, 2]$, and self-focusing or defocusing Kerr nonlinearity. Some fundamental solitons are obtained by means of the variational approximation, which are verified by comparison with numerical results. We find that the soliton collapse, exhibited by the one-dimensional cubic fractional nonlinear Schr\"{o}dinger equation with only one L\'{e}vy index $\alpha =1$, can be suppressed in the two-L\'{e}vy-index fractional nonlinear Schr\"{o}dinger system. Stability of the solitons is also explored against collisions with Gaussian pulses and adiabatic variation of the system parameters. Modulation instability of continuous waves is investigated in the two-L\'{e}vy-index system too. In particular, the modulation instability may occur in the case of the defocusing nonlinearity when two diffraction coefficients have opposite signs. Using results for the modulation instability, we produce first- and second-order rogue waves on top of continuous waves, for both signs of the Kerr nonlinearity. Show Chinese abstract

Abstract: 在具有两个分数色散/衍射项的激光系统中，这些项由它们的Lévy指数量化，$\alpha_{1}\, \alpha_{2}\in (1, 2]$，并具有自聚焦或自散焦的Kerr非线性。 一些基本孤子是通过变分近似方法获得的，并通过与数值结果的比较进行了验证。 我们发现，仅具有一个Lévy指数$\alpha =1$的一维立方分数非线性薛定谔方程表现出的孤子坍缩，在具有两个Lévy指数的分数非线性薛定谔系统中可以被抑制。 还研究了孤子在与高斯脉冲碰撞以及系统参数绝热变化下的稳定性。 连续波的调制不稳定性也在具有两个Lévy指数的系统中进行了研究。 特别是，当两个衍射系数符号相反时，在自散焦非线性情况下可能会发生调制不稳定性。 利用调制不稳定性结果，我们产生了在连续波之上的第一阶和第二阶异常波，适用于Kerr非线性的两种符号。