| dc.description |
We consider N-soliton solutions of the KP equation, 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<sup>+</sup>, n<sup>−</sup>) with n<sup>±</sup> = (n<sup>±</sup><sub>1</sub>, …, n<sup>±</sup><sub>N</sub>), n<sup>±</sup><sub>j</sub> ϵ {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<sup>+</sup>, n<sup>−</sup>). We also show that N-soliton solutions of the KdV equation obtained by the constraint ∂u/∂y = 0 cannot have resonant interaction. |
