Combinatorics and Geometry of KP solitons

Let $Gr(N,M)$ be the real Grassmann manifold defined by the set of all $N$-dimensional subspaces of $\mathbb{R}^M$. Each point on  $Gr(N,M)$ can be represented by an $N\times M$ matrix $A$ of rank $N$. If all the $N\times N$ minors of $A$ are nonnegative, the set of all points associated with those matrices forms the totally nonnegative part of the Grassmannian, denoted by $Gr(N,M)^+$.

In this talk, I start to give a realization of $Gr(N,M)^+$ in terms of the (regular) soliton solutions of the KP (Kadomtsev-Petviashvili) equation which is a two-dimensional extension of the KdV equation. The KP equation describes small amplitude and long waves on a surface of shallow water. I then construct a cellular decomposition of $Gr(N,M)^+$ with the asymptotic form of the soliton solutions. This leads to a classification theorem of all solitons solutions of the KP equation, showing that each soliton solution is uniquely parametrized by a derrangement of the symmetric group $S_M$. Each derangement defines a combinatorial object called the Le-diagram (a Young diagram with zeros in particular boxes). Then I show that the Le-diagram provides a complete classification of the "entire" spatial patterns of the soliton solutions coming from the $Gr(N,M)^+$ for asymptotic values of the time. I will also present some movies of real experiments of shallow water waves which represent some of those solutions obtained in the classification problem. If time permits, I will discuss an application of those results to analyze the Tohoku-tsunami of March 2011. The talk is elementary, and shows interesting connections among combinatorics, geometry and integrable systems.