Comment on the 3+1 dimensional Kadomtsev–Petviashvili equations

We comment on traveling wave solutions and rational solutions to the 3+1 dimensional Kadomtsev–Petviashvili (KP) equations: (ut + 6uux + uxxx)x ± 3uyy ± 3uzz = 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. 2010 Elsevier B.V. All rights reserved. The 3+1 dimensional Kadomtsev–Petviashvili (KP) equations read ðut þ 6uux þ uxxxÞx 3uyy 3uzz 1⁄4 0; ð1Þ which describe three-dimensional solitons in weakly dispersive media [1], particularly in fluid dynamics and plasma physics [2,3]. Like the 2 + 1 dimensional KP equations, the 3+1 dimensional KP equations with the negative sign ‘‘ ” and the positive sign ‘‘+” are called the 3+1 dimensional KP-I and KP-II equations, respectively. Several classes of exact traveling wave solutions to the 3+1 dimensional KP-II equation were presented by various authors (see, e.g., [4–6]). Very recently, traveling wave solutions and rational solutions were discussed and generated in [7,8] for the 3+1 dimensional KP-I equation, based on the homogeneous balance method and under help of a Riccati equation. In this note, on one hand, we would like to explain that more general traveling wave and rational solutions, including the ones presented in [4–8], can be constructed under transformations of dependent and independent variables. Three exact and explicit solutions to a Riccati equation will help in generating those solutions. This also contributes to identifying and correcting one common error in finding exact solutions to nonlinear wave equations: not using sufficiently general classes of solutions to ordinary differential equations [9]. On the other hand, using the Hirota bilinear method, we would like to analyze the existence condition of the three-soliton solution, and present the one-soliton and two-soliton solutions and four classes of specific three-soliton solutions to both of the 3+1 dimensional KP equations. It is known that many direct methods are nowadays available for constructing exact traveling wave solutions to nonlinear differential equations. The Riccati equation /n 1⁄4 a/ þ b ða – 0Þ ð2Þ plays a crucial role in manipulating nonlinear equations to get exact solutions by the homogeneous balance method. This equation has the following three exact solutions [10]: . All rights reserved. th.usf.edu 2664 W.X. Ma / Commun Nonlinear Sci Numer Simulat 16 (2011) 2663–2666 / 1⁄4 1 an þ n0 ; n0 1⁄4 const:; ð3Þ