In this article, we employ the perturbed Fokas-Lenells equation (FLE), which represents recent electronic communications. The Riccati-Bernoulli Sub-ODE method which does not depend on the balance rule is used for the first time to obtain the new exact and solitary wave solutions of this equation. This technique is direct, effective and reduces the large volume of calculations.

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In this article, we employ the Painlevé approach to realize the solitary wave solution to three distinct important equations for the shallow water derived from the generalized Camassa-Holm equation with periodic boundary conditions. The first one is the Camassa-Holm equation, which is the main source for the shallow water waves without hydrostatic pressure that describes the unidirectional propagation of waves at the free surface of shallow water under the influence of gravity. While the second, the Novikov equation as a new integrable equation, possesses a bi-Hamiltonian structure and an infinite sequence of conserved quantities. Finally, the third equation is the (3 + 1)-dimensional Kadomtsev-Petviashvili (KP) equation. All the ansatz methods with their modifications, whether they satisfy the balance rule or not, fail to construct the exact and solitary solutions to the first two models. Furthermore, the numerical solutions to these three equations have been constructed using the variational iteration method.

In this article the perturbed Gerdjikov-Ivanov (GI)-equation which acts for the dynamics of propagation of solitons is employed. The balanced modified extended tanh-function and the non-balanced Riccati-Bernoulli Sub-ODE methods are used for the first time to obtain the new optical solitons of this equation. The obtained results give an accuracy interpretation of the propagation of solitons. We held a comparison between our results and those are in the previous work. The efficiency of these methods for constructing the exact solutions has been demonstrated. It is shown that these different technique's reduces the large volume of calculations.