The computation of higher-order rogue waves can be a time-consuming process. Hirota bilinear equations with controllable centers for all orders of rogue wave solutions can be solved by a simplified symbolic computation method. Only one-third of the computational effort needed by the original method is required by the simplified one, for very high orders. For instance, simpler first- to fourth-order rogue waves, as well as fifth-order rogue waves that were previously unpublished, are yielded by the simplified method for a generalized (2+1)-dimensional Kadomtsev-Petviashvili equation. Moreover, the structural characteristics of the roots of auxiliary polynomials associated with higher-order rogue waves were systematically examined within the complex plane. In addition, a comprehensive analysis was conducted on their dynamic behavior by tracking the temporal evolution of the rogue wave profiles at the same time. To be honest, the obtained results in this paper substantially improve or complement the corresponding conditions in the known references. As a consequence, this paper gives a new idea to derive high-order rogue waves for the generalized (2+1)-dimensional Kadomtsev-Petviashvili equation.

The main attention of this paper focuses on studying the dynamical behaviors, cubic-quartic optical solitons, chaotic pattern and phase portraits for the nonlinear coupled Kaup-Newell equation in birefringent fibers. First of all, by means of traveling wave transforms and homogeneous balance principle, the coupled Kaup-Newell equation in birefringent fibers is simplified into ordinary differential equation. Secondly, the dynamical properties of two dimensional system and the corresponding perturbed system have been studied. Finally, with the assistance of the complete discriminant system method, the optical soliton solutions of the coupled Kaup-Newell equation in birefringent fibers can be derived, which include solitary wave solutions, ational function solutions, Jacobian elliptic function solutions and hyperbolic function solutions. In addition, two-dimensional portraits, three-dimensional portraits, contour plots, density plots and two-dimensional gradplots of the obtained solutions are also given by explaining the propagation of optical solitons for the coupled Kaup-Newell equation in birefringent fibers.

This work mainly concerned on the optical soliton solutions and chaotic pattern for the coupled stochastic Schrödinger-Hirota equation with multiplicative white noise in magneto-optic waveguides. Firstly, by means of traveling wave transformations and homogeneous balance principle, the the coupled stochastic Schrödinger-Hirota equation in magneto-optic waveguides is transformed into ordinary differential equation. By selecting some suitable parameters, phase diagrams are plotted with the help of the mathematical software Maple. Secondly, the optical soliton solutions of the coupled stochastic Schrödinger-Hirota equation corresponding to phase orbits can be easily deduced through the method of dynamical systems. Finally, the two-dimensional and three-dimensional graphs of the stochastic Schrödinger-Hirota equation are drawn, which further explain the propagation of the coupled stochastic Schrödinger-Hirota equation in nonlinear optics.