On the integrability of a generalized variable-coefficient Kadomtsev–Petviashvili equation

Abstract

By considering the inhomogeneities of media, a generalized variable-coefficient Kadomtsev–Petviashvili (vc-KP) equation is investigated, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. In this paper, we systematically investigate the complete integrability of the generalized vc-KP equation under an integrable constraint condition. With the aid of generalized Bell’s polynomials, its bilinear formalism, bilinear Bäcklund transformations, Lax pairs and Darboux covariant Lax pairs are succinctly constructed, which can be reduced to the ones of several integrable equations such as KdV, cylindrical KdV, KP, cylindrical KP, generalized cylindrical KP, non-isospectral KP equations, etc. Moreover, the infinite conservation laws of the equation are found by using its Lax equations. All conserved densities and fluxes are given with explicit recursion formulas. Furthermore, an extra auxiliary variable is introduced to obtain the bilinear formalism, based on which, the soliton solutions and Riemann theta function periodic wave solutions are presented. The influence of inhomogeneity coefficients on solitonic structures and interaction properties are discussed for physical interest and possible applications by some graphic analysis. Finally, a limiting procedure is presented to analyze in detail the asymptotic behavior of the periodic waves and the relations between the periodic wave solutions and soliton solutions.