Bilinear method and semi-inverse variational principle approach to the generalized (2+1)-dimensional shallow water wave equation

Abstract

In this article, we investigate the generalized (2+1)-dimensional shallow water wave equation which enables an unidirectional propagation of shallow-water waves. By noticing that the system is integrable, we could get the diverse forms of the solitary wave solutions by using the rogue wave and semi-inverse variational principle (SIVP) schemes. In particular, we investigate four solutions including rogue wave, soliton, bright soliton, dark soliton, and lump solutions. To achieve this, an illustrative example of the Helmholtz equation is provided to demonstrate the feasibility and reliability of the used procedure in this study. The effect the free parameters on the behavior of acquired figures to a few obtained solutions containing two nonlinear rational exact cases was also analyzed due to the nature of nonlinearities. The dynamic properties the obtained results are shown and analyzed by some density, two and three-dimensional images and also are presented the physical nature of obtained solutions.