We have investigated the propagation dynamics of super-Gaussian optical beams in fractional Schrödinger equation. We have identified the difference between the propagation dynamics of super-Gaussian beams and that of Gaussian beams. We show that, the linear propagation dynamics of the super-Gaussian beams with order m > 1 undergo an initial compression phase before they split into two sub-beams. The sub-beams with saddle shape separate each other and their interval increases linearly with propagation distance. In the nonlinear regime, the super-Gaussian beams evolve to become a single soliton, breathing soliton or soliton pair depending on the order of super-Gaussian beams, nonlinearity, as well as the Lévy index. In two dimensions, the linear evolution of super-Gaussian beams is similar to that for one dimension case, but the initial compression of the input super-Gaussian beams and the diffraction of the splitting beams are much stronger than that for one dimension case. While the nonlinear propagation of the super-Gaussian beams becomes much more unstable compared with that for the case of one dimension. Our results show the nonlinear effects can be tuned by varying the Lévy index in the fractional Schrödinger equation for a fixed input power.

We study both analytically and numerically the propagation dynamics of an initially chirped Airy pulse in an optical fiber. It is found that the linear propagation of an initially chirped Airy pulse depends considerably on whether the second-order dispersion parameter ${\beta }_2$ and chirp $C$ have the same or opposite signs. For ${\beta }_{2}C<0$, the chirped Airy pulse first undergoes an initial compression phase, then reaches a breakup area as depending on the values of $C$, and then experiences a lossy inversion transformation such that it continues to propagate with an opposite acceleration. The chirped Airy pulse is always dispersed during propagation in the case of ${\beta }_{2}C>0$. The impact of truncation coefficient and Kerr nonlinearity on the chirped Airy pulse propagation is also disclosed separately.

We investigate the mutual interaction of two spatially-separated Airy beams in the nonlinear Schrödinger equation with the fractional Laplacian. Depending on the beam separation ($d$), relative phase and Lévy index ($\alpha $), we observed an anomalous attraction or repulsion between the Airy beams. Anomalous attraction leads to a single breather soliton with a period that grows exponentially as $\alpha $ increases. In this region of the parameter space, we identify a crossover between two asymmetric regimes: as the Lévy index exceeds a critical value ${\alpha _c}$, the period of breather soliton for $d > 0$ is orders of magnitude larger than for $d < 0$, while the opposite occurs as $\alpha < {\alpha _c}$. Our results reveal a novel scenario for Airy beams interaction in the framework of fractional nonlinear Schrödinger equation and provide an alternative mechanism to control breather soliton generation.

Synthesis strategies, characterization methods, and various supports for Ru-based single-atom catalysts and their HER performance are reviewed.