The current work studies the bifurcation and the classification of single traveling wave solutions of the coupled version of Radhakrishnan-Kundu-Lakshmanan equation that usually describes the dynamics of optical pulses in fiber Bragg gratings, which is also described by a family of nonlinear Schrödinger equations with cubic nonlinear terms. The solutions of the hyperbolic functions, the rational functions, the trigonometric functions and the Jacobian functions are retrieved by using the complete discrimination system of polynomial. By selecting appropriate parameters, phase portraits, two-dimension graphics and three-dimension graphics of the obtained solutions are drawn.

This article studied the new traveling wave solutions of the cascaded model with higher-order dispersion effects combined with the effects of spatiotemporal dispersion and multiplicative white noise. In the process of exploring traveling wave solutions, a clever combination of the polynomial complete discriminant system was used to discover more forms of traveling wave solutions for this equation. In order to better observe and analyze the propagation characteristics of traveling wave solutions, we used Maple and Matlab software to provide two-dimensional and three-dimensional visualization displays of the equation solutions. Meanwhile, we also analyzed the internal mechanism of nonlinear partial differential equations using planar dynamical systems. The research results indicated that there are differences in the results of different forms of soliton solutions affected by external random factors, which provided more beneficial references for people to better understand the cascaded model with higher-order dispersion effects combined with the effects of spatiotemporal dispersion and multiplicative white noise, and helped people to more comprehensively understand the propagation characteristics of optical solitons. The solution method in this article was also applicable to the study of other nonlinear partial differential equations.