Diverse soliton wave profile analysis in ion-acoustic wave through an improved analytical approach

In engineering and applied sciences, several physical phenomena can be more precisely characterized by employing nonlinear fractional partial differential equations. The primary goal of this research is to examine the traveling wave solution in closed form for the nonlinear acoustic wave propagation model known as the time fractional simplified modified Camassa–Holm equation, which is used to explain the unidirectional propagation of shallow-water waves with non-hydrostatic pressure and explains the dispersion properties of numerous phenomena like fluid flow, control theory, liquid drop patterning in plasma, acoustics, fusion, and fission processes, etc. The utmost potential approach, namely the new auxiliary equation technique, is applied for analyzing the time nonlinear fractional simplified modified Camassa-Holm equation in the logic of the newest established truncated M-fractional derivative. The fractional partial differential equations have been transformed to the ordinary differential equation using the complex wave transformation in the sense of truncated M-fractional derivative. A variety of soliton solutions, including anti-kink, single soliton, anti-bell, bell, kink, multiple soliton, double soliton, singular-kink, compacton shape, periodic shape, and so many, are displayed in the diagram of 3D and contour plots by taking into account a number of various parameters. It is essential to point out that all derived outcomes are directly compared to the original solutions to certify their exactness. Results show that the used scheme is capable, simple, and straightforward and can be useful to a variety of complex phenomena. The acquired results are unique for the model equation and could be applied to the analysis of several nonlinear study fields.