Insights into the Vakhnenko–Parkes equation: solitary waves under the influence of power-law nonlinearity

Abstract

In this study, we identify two distinct families of solutions for the Vakhnenko–Parkes (VP) equation using a hybrid computational scheme. This methodology, referred to as the hybrid scheme, integrates B-spline functions with a finite-element approach. The notable advantage of this scheme is its unconditional stability, which enables the successful development of both topological and non-topological solutions. To demonstrate its efficacy, several test cases are examined. We visualise the dynamics of topological and non-topological solitary wave profiles by presenting results through figures and calculating the associated errors. Furthermore, we derive an analytical solution using the Khater II technique and compute conservation laws. In summary, our findings suggest that the presented scheme is highly effective and adaptable to various other nonlinear models.