Exact Solutions of the Schamel Korteweg-de Vries Equation byExtended Direct Algebraic Method

This study examines the analytical investigation of nonlinear wave structures governed by a Schamel-type Korteweg-de Vries (S-KdV) equation, essential in plasma physics for modelling ion-acoustic waves. This work is motivated by the necessity to enhance understanding of the impact of electron trapping effects on solitary wave dynamics. We employ the Extended Direct Algebraic (EDA) method to derive precise wave solutions. The applied method offers a systematic approach to derive various soliton structures and improves our comprehension of their physical properties. In plasma physics, the S-KdV equation is utilised to examine dust ion acoustic waves. Furthermore, it is employed to examine shallow water waves distinguished by steepening and breaking. It is applied in studying the Earth's magnetosphere, the solar wind, the nonlinear plasma turbulence, and the dusty space plasma. Graphical and comparative analyses are presented to validate the results and demonstrate the robustness of the method. The obtained solutions for the S-KdV equation have Kink type, anti-kink type, and multisoliton and solitary wave structures. The properties of the wave structures are demonstrated through the two-dimensional, three-dimensional, and contour plots. Additionally, the impact of the nonlinear term as well as the dispersion term on some of the obtained solutions are discussed through the two dimensional plots.