Phase portrait analysis and exact solutions of the stochastic complex Ginzburg-Landau equation with cubic-quintic-septic-nonic nonlinearity governing optical propagation in highly dispersive fibers

The study of the impact of random perturbations on exact solutions, such as soliton solutions, of nonlinear partial differential equations, is of great importance, as it helps establish the theoretical foundations necessary for practical applications. In this paper, we study further the problem of finding exact solutions of the complex Ginzburg-Landau equation incorporating eighth-order dispersion, high nonlinearity and random perturbation effect described by multiplicative white noise. We transform the problem of identifying exact solutions into the problem of solving an auxiliary second-order ordinary differential equation whose coefficients satisfy several algebraic equations simultaneously and which has a cubic polynomial as its nonlinearity, we utilize the phase portrait analysis method of Hamiltonian dynamical systems to conceptualize the structure of solutions to the auxiliary ordinary differential equations, and we borrow some idea of the method of complete discrimination systems for polynomials to calculate (the profiles of) the amplitude functions, thereby obtaining the desired exact solutions. To make our approach more accessible for practical use, we also provide clues for determining the coefficients of the auxiliary ordinary differential equation. In the meantime, we provide several numerical simulations to illustrate our theoretical results. Aside from the perturbed soliton solutions (corresponding to perturbed bright and dark optical solitons) reported in the existing literature, we find perturbed periodic and singular solutions in the concerned Ginzburg-Landau equation. The phase portrait analysis approach used in this paper helps to intuitively visualize the structure of exact solutions, thereby reducing the labor involved in calculating exact solutions to nonlinear partial differential equations. The Ginzburg-Landau equation can be utilized to portray the transmission of waves in optical metamaterials, and therefore, our theoretical research in this paper lays the groundwork for the identification of new solitons which are well-suited for optical communications and contributes to the advancement of optical technologies.