Symmetry methods and multi-structure solutions for a (3+1)-dimensional generalized nonlinear evolution equation

In this paper, we investigate a novel (3+1)-dimensional generalized Painlevè-integrable nonlinear evolution equation. Employing a dependent variable transformation, we derive the Hirota bilinear form, leading to the discovery of one, two, and three kink-soliton solutions for the equation. Furthermore, by substituting a quadratic-type test function into the Hirota bilinear form, we obtain lump solutions. Additionally, we extend our findings to include lump-multi-kink solutions using two distinct types of test functions. Furthermore, we establish two separate bilinear Bäcklund transformations using two different exchange identities, each characterized by its own set of arbitrary parameters. The first Bäcklund transformation form includes seven arbitrary parameters, while the second form features four arbitrary parameters. Our work also results in the discovery of a new exact traveling wave solution under various parametric conditions for our model. We delve into the dynamical behavior of these solutions, particularly in the long wave limit. Moreover, we explore the Lie point symmetries of our model equation, leading to the identification of new exact solutions arising from symmetry reduction.