Young Diagrams and N-soliton Solutions of the Kp Equation

We consider N-soliton solutions of the KP equation, (−4ut + uxxx + 6uux)x + 3uyy = 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 Nsoliton solutions include all possible resonant interactions among those line solitons. We then classify those N-soliton solutions by defining a pair of Nnumbers (n,n) with n = (n 1 , . . . , n N ), n±j ∈ {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.