In this study, a generalized form of the complex Ginzburg–Landau equation incorporating various nonlinear response
laws namely, the Kerr law, power law, parabolic, and dual-power nonlinearities is examined. By applying a suitable traveling wave
transformation, the governing partial differential equation is reduced to a nonlinear ordinary differential equation. After additional
variable shifts, the system is expressed in Hamiltonian form, from which the corresponding first integral is derived to analyze
the equilibrium structure. The qualitative behavior of the system is investigated by constructing phase portraits under different
parameter regimes, revealing the existence and stability characteristics of both singular and symmetric equilibria. Subsequently, the
modified Kudryashov method is employed to obtain several new classes of exact solutions for the considered nonlinear models.
These solutions are presented in hyperbolic, exponential, and rational function forms, corresponding to bright, dark, singular, kink,
and anti-kink wave profiles. To illustrate the physical features and propagation dynamics of the obtained solutions, two-dimensional
and three-dimensional graphical representations are plotted for appropriate parameter choices. The analytical and graphical results
demonstrate the effectiveness of the proposed approach in describing nonlinear wave evolution across diverse nonlinear optical
media.
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