Exact solutions and stochastic dynamics of the perturbed Chen–Lee–Liu equation with multiplicative white noise via a complete discriminant system

This paper investigates the dynamical behaviour and exact solutions of the generalized Chen-Lee-Liu (CLL) equation with multiplicative white noise. Firstly, the qualitative analysis of the equation subjected to the travelling wave transformation is performed by employing bifurcation theory, and the chaotic evolution mechanism of the system is revealed by introducing the perturbation term; the chaotic state is identified by means of Lyapunov exponents, and the sensitive dependence of the system on initial conditions is analysed. Based on this, the polynomial complete discrimination system developed by Liu, one of the proposers of the CLL equation, was employed to systematically solve the equation for the three cases where the nonlinear order parameter n=1,2,3. A rich set of exact solutions, including solitary wave solutions, periodic solutions, and Jacobi elliptic function solutions, was obtained. Two-dimensional and three-dimensional figures of typical solutions are plotted through numerical simulations, realizing the physical visualization of exact solutions and their stochastic averages. The research shows that the delay factor introduced by white noise causes the stochastic averages of the solutions to lose the inherent solitary and periodic characteristics, while their non-average counterparts completely retain these properties. The research framework of this paper combines chaos analysis, bifurcation theory, and the polynomial complete discrimination system, providing a new analytical perspective and theoretical basis for the dynamical research of the CLL equation under stochastic perturbations.