Exact solutions to the fractional complex Ginzburg–Landau equation with time-dependent coefficients under quadratic–cubic and power law nonlinearities

In this paper, the fractional complex Ginzburg–Landau equation with time-dependent coefficients, which is used to depict diverse physical phenomena like superfluidity, superconductivity, Bose–Einstein condensation, second-order phase transitions, strings and liquid crystals, is investigated by three different methods under the circumstance of taking two forms of nonlinearity into account. A number of exact solutions are derived and fallen into different categories for the sake of discussing the dynamic behaviors of this equation further. More narrowly, we acquire the solitary, soliton and elliptic wave solutions through the unified method as well as the trigonometric, hyperbolic trigonometric and rational solutions through the improved F-expansion method in the sense of quadratic–cubic nonlinearity. And as another achievement, we obtain the bright, dark, combined bright–dark, singular soliton, mixed singular soliton and singular periodic wave solutions by employing the extended Sinh-Gordon equation expansion method under power law nonlinearity. Moreover, we draw the 2D, 3D and contour images for some of these solutions with fit choices of parameters to realize the propagations of waves more visually and deeply, thereby uncovering more physical phenomena in connection with the governing model.