This article deals with the dynamical behavior of weakly nonlinear gravity waves propagating including the viscosity and surfactant on the ocean free surface. Through the bifurcation method, we could predict the nature of solutions of the nonlinear Schrödinger equation (NLSE) and reduce it to the nonlinear ordinary differential equation, easily solvable. Then, bright soliton, dark soliton, and Jacobi elliptic functions solutions of this NLSE under the viscosity and surfactant effect have been derived. These obtained results are central in hydrodynamics and can predict physical phenomena such as gravity wave propagation in deep water. Moreover, they allow to enhance the decision-making process and the acquisition of radar and lidar data on the ocean surface.