A study on the highly nonlinear Korteweg-de Vries–Kadomtsev-Petviashvili equation’s solitary wave solutions arises in fluids

Abstract

This article gives an exhaustive study of the sixth-order highly nonlinear Korteweg-de Vries–Kadomtsev-Petviashvili (KdV–KP) equation with emphasis on the development of analytical solutions and discussing its dynamical behavior. The study is carried out with two main goals. First, a wide range of exact solutions is obtained for certain parametric conditions through the improved modified extended tanh function method, a powerful analytical method suitable for dealing with intricate nonlinear wave equations. The obtained solution set contains various forms of dark, bright, and singular solitons, Jacobi elliptic function solutions, rational forms, exponential profiles, and singular periodic solutions. The analytic results not only enrich the catalog of admissible waveforms of the KdV–KP model but also offer closed-form benchmarks for verifying numerical simulations and further analytical studies. From an application standpoint, the derived solutions have significant relevance to physical contexts where strongly nonlinear wave interactions arise, such as shallow water dynamics, plasma physics, fluid mechanics, and coastal engineering. In particular, the physical nature of the obtained solutions may aid in designing and optimizing models for wave propagation in harbors, ports, and beach environments. Second, in order to have a better understanding of the qualitative properties of these solutions, we provide a large number of visualizations, such as two-dimensional profiles, three-dimensional surface plots, and contour maps, for some representative cases of solutions. Such graphical representations allow for easier comprehension of the spatial and temporal profiles, stability characteristics, and localization modes of the obtained waveforms. In total, the approach and findings of this research provide a comprehensive template for the investigation of higher-order nonlinear dispersive models, opening up possibilities for further theoretical advances and real-world applications in the field of wave dynamics research.