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

TL;DR

This paper investigates the complete integrability of a generalized variable-coefficient KP equation, constructing bilinear forms, Lax pairs, and solutions, revealing how inhomogeneities affect soliton and periodic wave behaviors.

Contribution

It systematically establishes integrability conditions, constructs bilinear forms, Lax pairs, and explicit solutions for the generalized vc-KP equation, extending understanding of inhomogeneous nonlinear wave equations.

Findings

Constructed bilinear formulism, Bäcklund transformations, and Lax pairs.

Derived infinite conservation laws with recursive formulas.

Presented soliton and periodic wave solutions, analyzing effects of inhomogeneity.

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 complete integrability of the generalized vc-KP equation under a integrable constraint condition. With the aid of a generalized Bells polynomials, its bilinear formulism, bilinear B\"{a}cklund 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 expressed in the form of accurate recursive formulas. Furthermore, an extra auxiliary variable is introduced to get the bilinear formulism, based on which, the soliton solutions and Riemann theta function periodic wave solutions are presented. And 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, asymptotic behavior of the periodic waves, and the relations between the periodic wave solutions and soliton solutions.