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Comment on the 3+1 dimensional Kadomtsev-Petviashvili equations. (English) Zbl 1221.35353
Summary: We comment on traveling wave solutions and rational solutions to the 3+1 dimensional Kadomtsev-Petviashvili (KP) equations: (u t +6uu x +u xxx ) x ±3u yy ±3u 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.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
35C07Traveling wave solutions of PDE
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
37K40Soliton theory, asymptotic behavior of solutions
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