Exploration of Soliton Solutions to the Special Korteweg–De Vries Equation with a Stability Analysis and Modulation Instability

This work is concerned with Hirota bilinear, expa function, and Sardar sub-equation methods to find the breather-wave, 1-Soliton, 2-Soliton, three-wave, and new periodic-wave results and some exact solitons of the special (1 + 1)-dimensional Korteweg–de Vries (KdV) equation. The model of concern is a partial differential equation that is used as a mathematical model of waves on shallow water surfaces. The results are attained as well as verified by Mathematica and Maple softwares. Some of the obtained solutions are represented in three-dimensional (3-D) and contour plots through the Mathematica tool. A stability analysis is performed to verify that the results are precise as well as accurate. Modulation instability is also performed for the steady-state solutions to the governing equation. The solutions are useful for the development of corresponding equations. This work shows that the methods used are simple and fruitful for investigating the results for other nonlinear partial differential models.