This study examines the analytic wave solutions of a highly dispersive perturbed complex Ginzburg–Landau equation (CGLE) with conformable fractional derivative and polynomial law of nonlinearity using the improved modified extended tanh-function method. The results show a wide range of solutions including (bright, dark, singular) solitons, Jacobi elliptic solutions, exponential solutions, and Weierstrass elliptic solutions. The obtained soliton solutions showcase diverse dynamics, encompassing different solitary waves and localized structures. The polynomial nonlinearity adds complexity to the dynamics, resulting in the emergence of new solitons with distinct characteristics. The impact of the fractional derivative is illustrated graphically using examples of some of the retrieved solutions with various values of fractional order.
