Young diagrams and N-soliton solutions of the KP equation

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 y --> infinity and y --> -infinity. 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(+), n(-)) with n(+/-) = (n(1)(+/-),...,n(N)(+/-)), n(j)(+/-) is an element of {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(+), n(-)). We also show that N-soliton solutions of the KdV equation obtained by the constraint partial derivative(u)/partial derivative(y) = 0 cannot have resonant interaction.