Triangulations and soliton graphs for totally positive Grassmannian

2018/08/05 by Karpman, Rachel, Kodama, Yuji
#Combinatorics (math.CO) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1808.01587

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 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.

Related

• KP solitons and total positivity for the Grassmannian
• Combinatorics of KP solitons from the real Grassmannian
• The Deodhar decomposition of the Grassmannian and the regularity of KP solitons
• KP theory, plabic networks in the disk and rational degenerations of M--curves
• Soliton solutions of the KP equation and application to shallow water waves