Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He's variational direct methods for the KP-BBM equation

This paper deals with cross-kink waves in the (2+1)-dimensional KP-BBM equation in the incompressible fluid. Based on Hirota's bilinear technique, cross-kink solutions related to KP-BBM equation are constructed. Taking the special reduction, the exact expression of different types of solutions comprising exponential, trigonometric and hyperbolic functions is obtained. Moreover, He's variational direct method (HVDM) based on the variational theory and Ritz-like method is employed to construct the abundant traveling wave solutions of the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation. These traveling wave solutions include kinky dark solitary wave solution, dark solitary wave solution, bright solitary wave solution, periodic wave solution and so on, which are all depending on the initial hypothesis for the Ritz-like method. In continuation, the modulation instability is engaged to discuss the stability of the obtained solutions. Moreover, the rational tan(theta/2) method on the generalized Hirota-Satsuma-Ito equation is investigated. The applicability and effectiveness of the acquired solutions are presented through the numerical results in the form of 3D and 2D graphs. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear waves.

Automated summary

This paper discusses the construction of different types of solutions in the KP-BBM equation and generalized Hirota-Satsuma-Ito equation in an incompressible fluid, as well as the stability and parameter influence of these solutions. The applicability and effectiveness of the solutions are demonstrated through numerical results and graphs, enriching the dynamic behavior of nonlinear wave evolution.