On a family of solutions of the KP equation which also satisfy the Toda lattice hierarchy
Abstract
We describe the interaction pattern in the
x
-
y
plane for a family of soliton solutions of the Kadomtsev-Petviashvili (KP) equation,
(−4
u
t
+
u
xxx
+6u
u
x
)
x
+3
u
yy
=0
. Those solutions also satisfy the finite Toda lattice hierarchy. We determine completely their asymptotic patterns for
y→±∞
, and we show that all the solutions (except the one-soliton solution) are of {\it resonant} type, consisting of arbitrary numbers of line solitons in both aymptotics; 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 {\it web-like} structure having
(
N
−
−1)(
N
+
−1)
holes.