Nonlinear Wave Features of the Time Fractional Gardner Equation Using Darboux Transformation

The focus of this article is to explore a class of multi-soliton nonlinear wave features of the time fractional Gardner (TFG) equation employing Darboux transformation with the aid of Lax pair. In the framework of this technique, first we set a transformation so that the TFG equation is converted into the standard Gardner equation and simultaneously the Lax pair of the foregoing equation can be achieved by employing Ablowitz-Kaup-Newell-Segur (AKNS) method. By determining Darboux transformation, multi-soliton nonlinear wave features of the TFG equation are constructed which allows us to encounter with new type of wave structures, like multi-kink type solitons, bell type soliton, compressive lump type soliton, breather-type soliton, etc. For better understanding two-dimensional and three-dimensional graphs are plotted of these nonlinear wave structures. Finally we will see by the effects of the fractional parameter how one structure is converted into another form.