Triangulations and soliton graphs for totally positive Grassmannian

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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)≥ 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.