We consider N-soliton solutions of the KP equation,
(-4u_t+u_{xxx}+6uu_x)_x+3u_{yy}=0 . An N-soliton solution is a solution
u(x,y,t) which has the same set of N line soliton solutions in both
asymptotics yββ and yβββ. The N-soliton solutions include
all possible resonant interactions among those line solitons. We then classify
those N-soliton solutions by defining a pair of N-numbers (n+,nβ) with nΒ±=(n1Β±β,...,nNΒ±β),njΒ±ββ{1,...,2N}, which labels N line solitons in the solution. The
classification is related to the Schubert decomposition of the Grassmann
manifolds Gr(N,2N), where the solution of the KP equation is defined as a
torus orbit. Then the interaction pattern of N-soliton solution can be
described by the pair of Young diagrams associated with (n+,nβ). We also show that N-soliton solutions of the KdV equation obtained by
the constraint βu/βy=0 cannot have resonant interaction.Comment: 22 pages, 5 figures, some minor corrections and added one section on
the KdV N-soliton solution