The 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 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) which labels N line solitons in the solution. The classification is related on the Schubert decomposition of the Grassmann manifolds, where the solution of the KP equation is defined as a torus orbit.