Integrability, breather, lump and quasi-periodic waves of non-autonomous Kadomtsev–Petviashvili equation based on Bell-polynomial approach

This article describes the infinite conservation law, quasi-periodic wave, breather, lump, and characteristic of integrability via bilinear Bäcklund and lax pairs of the non-autonomous Kadomtsev–Petviashvili (NKP) equation (in the presence of an external force and damping). Bell-polynomial is deduced from Bäcklund transformation from which infinite conservation law for NKP equation is derived. Obtaining well-known quasi-periodic solutions is an important aspect of this investigation. A limiting procedure is used for analyzing the asymptotic behavior of multi-periodic waves. This method proves rigorously that periodic waves tend toward soliton solutions under a small amplitude limit, which results in a relationship between periodic waves and soliton solutions. For the first time, the effect of external force in a quasi-periodic wave is demonstrated explicitly from the numerical understanding. Further, some complicated solution structures of the NKP equation such as lump and breather-type solitons are explored from the bilinear form of the said equation with the appropriate choice of polynomial functions. Important characteristics of lump and breather waves in different forcing backgrounds are graphically described. Finally, the lump wave of the NKP equation is explored from the breather wave in the limiting stage. It is also confirmed from the analytical results of the relevant motions that the velocity, maximum altitude, and interacting natures of the wave quantities are all influenced by the damping and forcing terms.