Exact soliton solutions, chaotic dynamics, and high-precision simulations for the modified complex Ginzburg–Landau equation

This study presents a unified framework for the modified complex Ginzburg–Landau (CGL) equation, addressing key gaps in traditional research, such as the neglect of coupling between fourth-order dispersion and Kerr nonlinearity, as well as improper damping term introduction. Following a coherent workflow from modeling to analytical solution, dynamical analysis, and high-precision numerical verification, we apply the generalized tanh function method combined with three Riccati equation solutions to derive eight classes of exact solutions, including periodic oscillatory waves, singular solitons, and dark solitons, covering specialized structures such as strongly localized spikes and symmetric dual peaks. Using bifurcation theory, Lyapunov exponent analysis, and return maps, we quantify the impact of external and stochastic perturbations on system evolution and soliton dynamics. In particular, this study focuses on the dynamic evolution characteristics of the system under noise perturbation and clearly reveals that noise perturbation can induce chaotic behavior in the system. For the first time, the Legendre–Gauss–Lobatto spectral method is employed for numerical verification, achieving accuracy below 2 × 10−8, strongly supporting the reliability of analytical solutions. This work advances the dynamical analysis of the modified CGL equation, addresses limitations of existing methods, and offers a theoretically robust and practically valuable reference for nonlinear optics and optical communications. The proposed unified framework also holds significance for similar nonlinear PDEs.