The KP equation is a two-dimensional non-linear differential equation, which provides an excellent model for resonant interactions among standing-water waves. Kodama and Williams showed a remarkable relationship between soliton graphs, which encode the asymptotic behavior of line soliton solutions of the KP equation, and Postnikov's combinatorial theory of the totally nonnegative Grassmannian. Line soliton solutions correspond to points in the Grassmannian, and soliton graphs which come from the totally positive Grassmannian turn out to be examples of Postnikov's plabic graphs. In addition, Kodama and Williams showed that soliton graphs corresponding to the totally positive part of the Grassmannian of planes have a nice combinatorial interpretation interms of triangulations. In this talk, we use triangulations to encode soliton graphs for the totally positive part of Gr(k,n) for all k. In the process, we recover Kodama and Williams' result that these soliton graphs are in fact plabic graphs. This project is joint with Yuji Kodama and Jihui Huang.
Speaker(s): Rachel Karpman (Ohio State University)