On lump, travelling wave solutions and the stability analysis for the (3+1)-dimensional nonlinear fractional generalized shallow water wave model in fluids

The shallow water equation, which describes fluid flow below a pressure surface, is the structure that is most frequently used for displaying free surface flows. This particular kind of approach takes into consideration the impact of gravity on fluids through the inclusion of a source term proportional to the product of the water level and ground slopes. The given model is converted into a bilinear form using the Hirota bilinear method. Which refers to the development of lump waves, collisions between lump waves and periodic waves, collisions between lump waves and single- and double-kink soliton solutions, and collisions between lump, periodic, and single- and double-kink soliton solutions. Furthermore, the exp\((-\varrho (\varsigma ))\) approach are applied to obtain several forms of innovative combinations for the governing dynamical fractional model. In addition, it has been confirmed that the established results are stable, and it has been helpful to validate the calculations. Moreover, multiple intriguing exact solutions are utilized to illustrate the physical nature of 3D, contour, and 2D graphs. The solution yields a set of bright, dark, periodic, rational, and elliptic function solutions.