Zusammenfassung
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 $y\toınfty$ and $y-ınfty$. 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 $(n^+,\bf
n^-)$ with $n^\pm=(n_1^\pm,...,n_N^\pm),
n_j^\pmın\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 $(n^+,\bf
n^-)$. We also show that $N$-soliton solutions of the KdV equation obtained by
the constraint $u/y=0$ cannot have resonant interaction.
Beschreibung
Young diagrams and N-soliton solutions of the KP equation
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