The Painlevé integrability and abundant analytical solutions for the potential Kadomtsev–Petviashvili (pKP) type coupled system with variable coefficients arising in nonlinear physics

Abstract

Acknowledging the inhomogeneities of media and non-uniformity of boundaries, the nonlinear evolution equations with variable coefficients may show more realistic situations dealing with time-varying environment, inhomogeneous medium, etc. In this order, this article deals with the variable-coefficients (2+1)-dimensional potential Kadomtsev–Petviashvili (pKP) type coupled equations. The Painlevé analysis approach is being used to assess the integrability of the governing model. The Laurent series, which appears when using the Painlevé analysis technique, has been truncated in order to build an auto-Bäcklund transformation (ABT). Three different analytical solutions in the form of hyperbolic, exponential, periodic and rational have been evaluated using the ABT approach. The multi-soliton solutions are also derived by using Hirota bilinear method for the aforesaid equation. All the solutions are displayed as 3D plots in which the variable coefficients and parameters can be varied to see the effects. In these plots, one can see the many different behaviors of the examined coupled system, such as periodic waves, kink-soliton, anti-kink soliton, kink anti-kink wave, complex periodic waves, complex kink anti-kink wave, and multi-soliton surfaces.