%0 Generic %1 Kodama2004 %A Kodama, Yuji %D 2004 %K solitons %T Young diagrams and N-soliton solutions of the KP equation %U http://arxiv.org/abs/nlin/0406033 %X 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.

@misc{Kodama2004, abstract = { 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 $y\to\infty$ and $y\to -\infty$. 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 $({\bf n}^+,{\bf n}^-)$ with ${\bf n}^{\pm}=(n_1^{\pm},...,n_N^{\pm}), n_j^{\pm}\in\{1,...,2N\}$, 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 $({\bf n}^+,{\bf n}^-)$. We also show that $N$-soliton solutions of the KdV equation obtained by the constraint $\partial u/\partial y=0$ cannot have resonant interaction. }, added-at = {2011-09-27T07:24:14.000+0200}, author = {Kodama, Yuji}, biburl = {https://www.bibsonomy.org/bibtex/2696ca41fa8a5cd04dba92312be175c9d/casvada}, description = {Young diagrams and N-soliton solutions of the KP equation}, interhash = {0a74ca6b290c2bf77649a6d1482fbd34}, intrahash = {696ca41fa8a5cd04dba92312be175c9d}, keywords = {solitons}, note = {cite arxiv:nlin/0406033Comment: 22 pages, 5 figures, some minor corrections and added one section on the KdV N-soliton solutions}, timestamp = {2011-09-27T07:24:14.000+0200}, title = {Young diagrams and N-soliton solutions of the KP equation}, url = {http://arxiv.org/abs/nlin/0406033}, year = 2004 }