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 → ∞ and y → -∞. 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 epsi {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 ∂u/∂y = 0 cannot have resonant interaction.