Integrability aspects, rational type solutions and invariant solutions of an extended (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

In this article, we examine an extended (3+1)-dimensional B-type Kadomtsev–Petviashvili equation, applicable for the analysis of various physical phenomena such as surface waves in water, plasma physics and nonlinear optics. We systematically explore the integrable properties of the nonlinear evolution equation under consideration from various perspectives. Firstly, we outline fundamental properties of multi-dimensional Bell polynomial theory and its connection with the Hirota bilinear form. Utilizing these relations, we derive the Hirota bilinear form and a bilinear Bäcklund transformation. Through the application of the Cole-Hopf transformation within the bilinear Bäcklund transformation, we establish a Lax pair. Additionally, employing Bell polynomial theory, we compute an infinite number of conservation laws. Furthermore, we explicitly obtain one- and two-soliton solutions from the Hirota bilinear form and represent them graphically. We analyze Hirota’s condition for three-soliton solution and demonstrate that our considered model does not satisfy Hirota’s three-soliton condition automatically. Moreover, the Lie symmetry approach is employed to investigate the Lie symmetries and vector fields embedded in the addressed problem. Subsequently, symmetry reductions are achieved through the utilization of similarity variables, leading to the acquisition of several closed-form solutions. These closed-form solutions provide valuable insights into the behavior of the problem and allow for a deeper understanding of its underlying dynamics. Further, first and second order breather solutions are also derived by incorporating appropriate complex conjugate parameters in the two-soliton solution and four-soliton solution respectively. We provide a density plot, contour plot of the breather solutions and illustrate the evolution of breather solutions using a 3D plot. Additionally, a breather-kink solution is derived by selecting suitable complex conjugate parameters in N-soliton solution and their interaction dynamics are depicted visually. Utilizing the Wronskian technique for the rational solution of the KdV equation, we successfully obtained the Wronskian rational solution for the considered model.