On a family of solutions of the Kadomtsev–Petviashvili equation which also satisfy the Toda lattice hierarchy

We describe the interaction pattern in the x–y plane for a family of soliton solutions of the Kadomtsev–Petviashvili (KP) equation, (−4ut + uxxx + 6uux)x + 3uyy = 0. The solutions considered also satisfy the finite Toda lattice hierarchy. We determine completely their asymptotic patterns for y → ±∞, and we show that all the solutions (except the 1-soliton solution) are of resonant type, consisting of arbitrary numbers of line solitons in both asymptotics; that is, arbitrary N− incoming solitons for y → −∞ interact to form arbitrary N+ outgoing solitons for y → ∞. We also discuss the interaction process of those solitons, and show that the resonant interaction creates a web-like structure having (N− − 1)(N+ − 1) holes. PACS numbers: 02.30.Jr, 02.30.Ik, 05.45.Yv (Some figures in this article are in colour only in the electronic version)