Painlevé integrability of a (3+1)-dimensional Hirota-type equation: multi-solitons, lumps, and other physical solutions

In this work, a new multi-dimensional evolution equation known as a (3+1)-dimensional Hirota-type equation is built and analyzed using several analytical schemes. Before proceeding to derive its various solutions, we perform the Painleve test and confirm its complete integrability with the help of the Mathematica package. We obtain the dispersion relations and phase shifts associated with the multiple solitons after the interaction using Hirota’s approach. In addition, we provide a range of lump solutions by utilizing symbolic computation utilizing Maple. Also, other exact solutions for various structures, including kink, periodic, single, and rational, will be derived using other approaches. To gain a deeper understanding of the properties and structure of the solutions, we create 3D- and 2D-dimensional representations of them using appropriate values for the associated parameters. These findings help to explain various nonlinear phenomena that might occur in the physics of plasma and optical fibers.