Prediction and dynamical evolution of multipole soliton families in fractional Schrödinger equation with the PT-symmetric potential and saturable nonlinearity

The first good prediction of the multipole soliton solution for the non-integrable equation, i.e., the saturable nonlinear Schrödinger equation under the PT-symmetric potential, is achieved using the physical information neural network. In addition, we construct multipole (tripole to sextupole) soliton families in saturable nonlinear media with fractional diffraction under the PT-symmetric potential, and quadrupole, pentapole and sextupole solitons can coexist for the same parameters. The existence of multipole solitons is modulated by the modulation intensity of the PT-symmetric potential and Lévy index altogether, while the stable domain of multipole solitons is modulated by both the power and Lévy index together. With the increase in the modulation intensity of the PT-symmetric potential and Lévy index, the existence domain of multipole solitons gradually enlarges. When the soliton power is conserved, with the add of the Lévy index, the peak amplitudes at the outermost part of the profiles of real and imaginary parts for the multipole soliton increase, while the peak amplitudes at other positions decrease, and yet the soliton width increases. In addition, the strong saturable nonlinearity not only reduces the stability of tripole solitons but also inhibits the instability of quadrupole and pentapole solitons. However, the saturable nonlinear intensity exists a threshold for the stability modulation of sextupole solitons, beyond which the stability of sextupole solitons is no longer modulated by the saturable nonlinearity.