A complete study of the Wick-type stochastic complex Ginzburg–Landau equation

In this study, we investigate the Wick-type stochastic complex Ginzburg–Landau equation (CGLE) with five distinct forms of nonlinear refractive indices. The Hermite transform is utilized to convert the Wick-type stochastic CGLE into a deterministic equation. Then the integral form is constructed by the traveling wave transform. Through the trial function approach, we demonstrate that the five forms of the Wick-type stochastic CGLE can be reduced into two primary forms for further discussion: the Kerr law and the polynomial law, with the other three forms derived analogously. When the coefficients are constant, we discuss the Hamiltonian and dynamic behaviors in detail, followed by an in-depth analysis of the chaotic behaviors. Furthermore, the traveling wave solutions with constant coefficients and the non-traveling wave solutions with variable coefficients are obtained using the complete discrimination system for polynomial method (CDSPM).