Shallow-water waves through two new generalized multi-dimensional variable coefficient equations

Abstract

The objective of this study is to propose and investigate two new forms of generalized variable coefficients within multi-dimensional equations describing shallow-water waves. We employ the Mathematica program to rigorously establish Painlevé's integrability for these two nonlinear equations. Subsequently, we constructed their bilinear forms and utilized Hirota's bilinear method to examine the dispersion relations and phase shifts of these two models that enable the derivative of multi-soliton solutions. Furthermore, diverse forms of lump-wave solutions are also considered. To illustrate the physical characteristics of these two models, we establish several graphical representations of the discovered solutions. These visualizations offer insights into the behavior, shape, and dynamics of both the multi-soliton, Peregrine soliton, lump wave, and rogue wave, enhancing our understanding of their physical significance. The two soliton solutions effectively replicate the shallow water waves, encompassing the T-, X-, and Y-types, along with other intricate interactions. Additionally, the lump and rogue wave structures are displayed to visually represent their spatial structures. These graphical representations offer a comprehensive view of the diverse wave phenomena observed in shallow water systems, aiding in the understanding of their spatial characteristics and interactions. Therefore, our findings indicate that the introduction of the two newly proposed integrable nonlinear evolution equations enhances the repertoire of integrable system models and aids in comprehending the distinctive characteristics of nonlinear dynamics in real-world applications.