Retrieval of diverse soliton, lump solutions to a dynamical system of the nonlinear (4+1) Fokas equation and stability analysis

This paper reveals soliton solutions to (4+1)-dimensional Fokas equation, which is an integrable extension of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations. In wave theory, the Fokas equation plays a crucial role in describing the physical phenomena of waves on the water's surface and beneath it. The observed model is subjected to an extended simple equation method (ESEM) and the Hirota bilinear method (HBM), which disclosed an abundance of soliton solutions in distinct formats, in the form of trigonometric functions, singular, periodic, rational, and exponential solutions. Moreover, we developed a number of solutions, such as the homoclinic breather wave solution,the periodic wave solution, the M-shaped rational wave solution, and the kink with their interaction solution, which are not documented in the literature. Additionally, modulation instability is effectively discussed. Some of the achieved results are explained in 2D, 3D, contour and density graphs. The new results interpreting that these obtained solutions can be a part, to complete the family of solutions and considered methods are effective, simple, and easy to use.