Exact analytical solutions of the modified complex Ginzburg-Landau equation: Multi-waves, bright and dark solitons via systematic integration approaches

In this study, we present the derivation of exact solutions for the modified complex Ginzburg–Landau (mCGL) equation. This equation is one of the models in plasmas, optical systems and spatially extended nonequilibrium media. Pursuing the Kumar–Malik ansatz, multi-wave expansion and dynamical phase-space analysis, we systematically construct many families of solutions. The Kumar–Malik method furnishes Jacobi elliptic solutions and hyperbolic and trigonometric forms. In this regard, we capture soliton solutions, including singular periodic, bright, dark, kink, anti-kink and singular waveforms. On the other hand, the multi-wave approach presents a rational solution that defines wave interaction, while dynamical analysis reveals bright and dark solitons through Hamiltonian phase portraits. 3-D, density and time plot profiles demonstrate the physical meanings of the exact solutions. With the examined analytic techniques, other nonlinear phenomena in diverse physical systems, such as optical communications, Bose–Einstein condensates and ultrafast laser systems can be investigated.