Dynamical analysis and exact soliton solutions of the truncated M-fractional Gardner–Kawahara model

This research reveals the novel types of exact wave solutions of the nonlinear Gardner–Kawahara (G–K) model in the concept of truncated M-fractional derivative. The G-K model, which is also called the extended Korteweg–de Vries (KdV) model, explains the solitary wave propagation in media, notation in plasmas, notation in shallow-water waves along surface tension and notation of magneto-acoustic waves. For our purpose, two techniques, the unified and the Sardar sub-equation techniques are applied. As a result, new types of exact wave solitons having periodic, dark–bright, periodic, kink are obtained. Some of the obtained solutions are represented through two- and three-dimensional and contour plots. The effect of the truncated M-fractional derivative (TMFD) is explained by plots. Stability of a concerned equation is checked by applying stability analysis. Moreover, the modulation instability analysis of the governing equation is also performed, which proves that the model and the obtained results are stable as well as exact.