Spatiotemporal propagation characteristics based on the exact solutions of the generalized nonlinear Schrödinger equation by intensity moments

Based on the exact solutions of the (3 + 1)-dimensional ((3 + 1)D) generalized nonlinear Schrödinger equation (GNLSE), we analyze the spatiotemporal propagation characteristics by intensity moments when a laser propagates in an inhomogeneous nonlinear medium. The different order intensity moment can describe the characteristics of a laser, and in the paper, the beam width (BW), the pulse width (PW), the skewness and the kurtosis parameter are calculated. The spatiotemporal propagation stability of the exact solutions is analyzed in detail by the second-order intensity moment. We find that when the diffraction and dispersion coefficients are the identical distributed functions, the BW and PW of the exact solutions are constants or vary periodically during nonlinear propagation. So, the spatiotemporal propagation of the exact solutions is stable. When the diffraction and dispersion coefficients are other coefficients, the BW and PW of the exact solutions vary irregularly due to the effect of a chirp. Thus the spatiotemporal propagation of the exact solutions is unstable. The results are helpful to the extendable investigation of nonlinear propagation and control of a laser pulse.