The University of Michigan Combinatorics Seminar
Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that the Wronskian approach to the KP equation provides a method to construct soliton solutions. More recently, several authors have focused on understanding the regular soliton solutions that one obtains in this way: these come from points of the totally non-negative (TNN) part of the Grassmannian.
In joint work with Yuji Kodama, we establish a tight connection between Postnikov's theory of total positivity for the Grassmannian, and the structure of regular soliton solutions to the KP equation. This connection allows us to apply machinery from total positivity to KP solitons. In particular, we completely classify the spatial patterns of the soliton solutions (which we call soliton graphs), coming from the totally positive (TP) part of Gr(2,n), as well as those coming from the TNN part of Gr(k,n), when the absolute value of the time parameter is sufficiently large. We also demonstrate an intriguing connection between soliton graphs for the TP part of Gr(k,n) and the cluster algebras of Fomin and Zelevinsky. We use this connection to solve the inverse problem for KP solitons coming from the TP part of Gr(k,n).