Abstract
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 GrassmannianGr(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 GrassmannianGr(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.
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