Painlevé Analysis and Determinant Solutions of a (3+1)-Dimensional Variable-Coefficient Kadomtsev–Petviashvili Equation in Wronskian and Grammian Form

Abstract

In this paper, the investigation is focused on a (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili (vcKP) equation, which can describe the realistic nonlinear phenomena in the fluid dynamics and plasma in three spatial dimensions. In order to study the integrability property of such an equation, the Painlevé analysis is performed on it. And then, based on the truncated Painlevé expansion, the bilinear form of the (3+1)-dimensional vcKP equation is obtained under certain coefficients constraint, and its solution in the Wronskian determinant form is constructed and verified by virtue of the Wronskian technique. Besides the Wronskian determinant solution, it is shown that the (3+1)-dimensional vcKP equation also possesses a solution in the form of the Grammian determinant.

Investigation on a nonisospectral fifth-order Korteweg-de Vries equation generalized from fluids

“…[32][33][34][35][42][43][44][45] There exist two ways in giving the constraint conditions among the coefficients, i.e., the Painlevé analysis 58 and process mapping the variablecoefficient models to the completely integrable constant-coefficient counterparts. 45,59 For some variable-coefficient NLEEs, Refs.…”

“…[21][22][23][24][25][26] Inhomogeneous media and boundaries can lead to the variable coefficients of the nonlinear evolution equations (NLEEs). [9][10][11][12][13][14][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] Generally speaking, it is easier to solve analytically the NLEEs with the scaled "temporal" variable coefficients than those with the scaled "spatial" ones from the viewpoint of integrability, just as those shown in Refs. [42][43][44][45][46].…”

A new N-soliton solutions for a generalized mKdV equation with variable coefficients

“…Since there are choices for the parameters, the variable coefficient KdV-typed equations can be considered as generalizations of the constant coefficient ones. Under certain constraint conditions, obtained by the Painlevé analysis [11,12,15,25,26,27,28] and conditions from the variable coefficient models mapped to the completely integrable constant coefficient counterparts [12,18], the variable coefficient KdV models may be proved to be integrable and have N -soliton solutions [3]. The corresponding constraint conditions on Eq.…”