Comparison Between the Exact Solutions of Three Distinct Shallow Water Equations Using the Painlevé Approach and Its Numerical Solutions

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.