In this paper, we study the Fokas–Lenells (FL) equation in the presence of perturbation terms and we construct exact solutions. The modified extended direct algebraic method (MEDAM) is utilized, and soliton solutions, solitary wave solutions and elliptic function solutions are achieved. The physical meaning of the geometrical shapes for some of the obtained results is investigated for various choices of the free parameters that appear in the results. The stability of the model is investigated by using the standard linear stability analysis, which verifies that the exact solutions are stable. The achieved solutions and computational work show that the current method is powerful and influential.