The KP equation is a nonlinear dispersive wave equation which provides an excellent model for resonant interactions of shallow-water waves. It is well known that regular soliton solutions of the KP equation may be constructed from points in the totally nonnegative Grassmannian Gr$(N,M)_{\geq 0}$. Kodama and Williams studied the asymptotic patterns (tropical limit) of KP solitons, called soliton graphs, and showed that they correspond to Postnikov's Le-diagrams. In this paper, we consider soliton graphs for the KP hierarchy, a family of commuting flows which are compatible with the KP equation. For the positive Grassmannian Gr$(2,M)_{>0}$, Kodama and Williams showed that soliton graphs are in bijection with triangulations of the $M$-gon. We extend this result to Gr$(N,M)_{>0}$ when $N=3$ and $M=6,7$ and $8$. In each case, we show that soliton graphs are in bijection with Postnikov's plabic graphs, which generalize Le-diagrams.

中文翻译：

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完全正 Grassmannian 的三角剖分和孤子图

KP 方程是一个非线性色散波动方程，它为浅水波的共振相互作用提供了一个很好的模型。众所周知，KP 方程的正则孤子解可以由完全非负的 Grassmannian Gr$(N,M)_{\geq 0}$ 中的点构成。Kodama 和 Williams 研究了 KP 孤子的渐近模式（热带极限），称为孤子图，并表明它们对应于 Postnikov 的 Le 图。在本文中，我们考虑 KP 层次结构的孤子图，这是一个与 KP 方程兼容的通勤流族。对于正 Grassmannian Gr$(2,M)_{>0}$，Kodama 和 Williams 表明孤子图与 $M$-gon 的三角剖分成双射。当 $N=3$ 和 $M=6,7$ 和 $8$ 时，我们将此结果扩展到 Gr$(N,M)_{>0}$。在每种情况下，