Triangulations and soliton graphs for totally positive Grassmannian

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