The control for multiple kinds of solitons generated in the nonlinear fractional Schro<spacing diaeresis>dinger optical system based on Hermite-Gaussian beams

In this paper, we investigate the control for Hermite-Gaussian (HG) solitons in the nonlinear fractional Schro<spacing diaeresis>dinger equation (FSE) by sequentially applying power function modulations, cosine modulations, parabolic potentials, and quadratic phase modulations (QPM). In the photorefractive media, the HG beam forms scattered breathing solitons when the fractional diffraction effect equilibrates with nonlinear effect. Under the power function modulation, the soliton maintains an equidistant linear transmission along the z-axis, and the number of solitons is equal to the mode. In the cosine modulation, the soliton distorts and its energy rapidly decreases after a certain distance of transmission. The time of the distortion varies with the Le<acute accent>vy index, photorefractive coefficient, modulation frequency and order. The freak spots exhibit a "flower" shape pattern. If a parabolic potential is introduced, the HG beam forms crawling soliton pairs or merges into a single bounded breathing soliton by adjusting the correlation among the Le<acute accent>vy index, nonlinear and parabolic coefficients. By increasing the nonlinear coefficient in the negative QPM regime, the defocusing HG beam emits several "filiform" breathing solitons during its propagation, which move in a parallel straight line to each other. The HG beam is transformed into a single fine breathing soliton after being focused under a positive QPM. The time of the formation and breathing rate varies with the Le<acute accent>vy index, QPM and nonlinear coefficients. Moreover, the number of solitons changes irregularly with modes. These results are significant for applications in optical communication, optical device design, and optical signal processing.