Comment on the 3+1 dimensional Kadomtsev–Petviashvili equations

Abstract

We comment on traveling wave solutions and rational solutions to the 3+1 dimensional Kadomtsev–Petviashvili (KP) equations: ( u t + 6 uu x + u xxx ) x ± 3 u yy ± 3 u zz = 0. We also show that both of the 3+1 dimensional KP equations do not possess the three-soliton solution. This suggests that none of the 3+1 dimensional KP equations should be integrable, and partially explains why they do not pass the Painlevé test. As by-products, the one-soliton and two-soliton solutions and four classes of specific three-soliton solutions are explicitly presented.