Analyzing specific waves and various dynamics of multi-peakons in (3+1)-dimensional p-type equation using a newly created methodology

In this article, we are proposing a newly created methodology known as the generalized exponential differential function method. By taking advantage of this method, we have constructed a trial solution of the reduced ordinary differential equation, which involves the ith derivative of the exponential rational function, which depends on the balancing of the equation. We introduce the generalized exponential differential function method to extract novel exact solutions for the (3+1)-dimensional p-type equation. To enhance the clarity of these solutions, we present 3-dimensional and contour plots illustrating the obtained solutions. Our visual representations reveal the presence of distinct features in the solutions, including lumps, solitons, multi-peakons, and interactions between solitons and waves. These solutions have a wide range of applications, including physics, engineering, plasma physics, ocean physics, nonlinear dynamics, and so on.